%% ==================================================================
%%  Exact orders and invariant anticanonical linear systems
%%  A self-contained account, rewritten to be as clear, correct,
%%  concise, approachable and beautiful as we know how to make it.
%%  The mathematical content is faithful to the original paper;
%%  every input from a companion preprint is named explicitly, so a
%%  reader who wants a fully closed proof can chase those down.
%% ==================================================================
\documentclass[11pt,reqno]{amsart}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern,microtype,amsmath,amssymb,mathtools,mathrsfs}
\usepackage[margin=1.08in]{geometry}
\usepackage{enumitem}
\usepackage[hyphens]{url}
\usepackage{xcolor}
\usepackage[unicode,colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}\newtheorem{remark}[theorem]{Remark}
\newcommand{\C}{\mathbb C}\newcommand{\Q}{\mathbb Q}
\newcommand{\R}{\mathbb R}\newcommand{\N}{\mathbb N}\newcommand{\Z}{\mathbb Z}
\newcommand{\OO}{\mathcal O}\newcommand{\PP}{\mathbb P}
\newcommand{\ddc}{i\partial\bar\partial}
\DeclareMathOperator{\divisor}{div}\DeclareMathOperator{\ord}{ord}
\DeclareMathOperator{\Spec}{Spec}\DeclareMathOperator{\coeff}{coeff}
\DeclareMathOperator{\rank}{rank}\DeclareMathOperator{\Pic}{Pic}
\DeclareMathOperator{\Supp}{Supp}\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\trdeg}{trdeg}\DeclareMathOperator{\codim}{codim}

\title[Exact orders and invariant anticanonical systems]%
      {Exact orders and invariant anticanonical linear systems}
\author{OpenAI}
\date{September 26, 2026}

\begin{document}
\raggedbottom
\maketitle

\begin{abstract}
On a smooth connected projective complex variety with a smoothly
semipositive anticanonical bundle, we prove that asymptotically zero
valuation on the curvature-null face is attained by an effective
rational anticanonical divisor.  The fixed auxiliary line bundle is
arbitrary.  Under an additional cohomological vanishing hypothesis we
obtain torus-invariant anticanonical sections, and two independent
descents give a scalar nonvanishing theorem for the anticanonical
bundle.
\end{abstract}

\section*{A note to the reader}
\addcontentsline{toc}{section}{A note to the reader}

This paper is written so that a curious high-school student can read
every paragraph, and a working algebraic geometer can verify every
line.  The trick is to separate three kinds of text.

\begin{itemize}[leftmargin=2em]
\item Definitions and basic facts are collected once, in
      Section~\ref{sec:conventions}.  If you have not met a term
      before, look there first.
\item Each major result begins with an \emph{In words} paragraph.  That
      paragraph is not a proof; it is the plain story.  The proof that
      follows is the real thing.
\item Long technical arguments cite a fixed list of companion preprints
      (labelled Companion E, G, H, I) and published theorems.  These are
      inputs.  If you want a completely closed proof you must also read
      those inputs.  Everything else here is proved from scratch.
\end{itemize}

Read the table of contents, then Section~\ref{sec:intro}, then jump
straight to the theorem you care about.  The paper is long only because
there are two completely different finishing moves, and a reader who
wants to understand either one should see it done carefully.

\tableofcontents

\section{Introduction}
\label{sec:intro}

\subsection{The central question}

Let $X$ be a smooth connected projective complex variety of dimension
$n>0$, and let
\[
  L \;=\; -K_X \;=\; \wedge^{n} T_X
\]
be its anticanonical line bundle: the top exterior power of the
tangent bundle.  A \emph{section} of the positive multiple $mL$ is,
concretely, a global holomorphic $n$-form with coefficients allowed to
have poles of the shape forced by $m$ (or, equivalently, with
coefficients localised by the inverse of $m$).  It is a global
object; it either exists or it does not.

The \emph{nonvanishing problem for the anticanonical bundle} asks
whether $H^0(X,mL)\ne 0$ for some integer $m>0$.  Restated, does the
anticanonical bundle ever have a nonzero section in a positive power?
When $X$ is a smooth compact surface with $-K_X$ semipositive
(equivalently $c_1(X)\le 0$), the answer is yes, but the reason is a
classical structure theory.  In higher dimension the question is open
in general and is the subject of a large recent literature.

Our paper proves a sharp answer to a related question: not merely
whether \emph{some} section exists, but whether one can force the
section to be nonzero at a prescribed center, using only rate
information about an auxiliary family of divisors.  This is stronger
and, as we will explain, more useful for anticanonical problems.

\subsection{Main results}

Let $\alpha$ be a smooth Hermitian curvature form on $L$ that is
semipositive (i.e.\ $\alpha\ge 0$ pointwise) and represents $c_1(L)$.
Let $\nu$ be the maximum complex rank of $\alpha$.  Since $\alpha\ge
0$, the top wedge $\alpha^{\nu+1}$ vanishes identically.  Fix an
ample line bundle $G_0$ and define
\begin{equation}\label{eq:ell}
  \ell(D) \;=\;
  \begin{cases}
    D\cdot L^\nu\cdot G_0^{\,n-\nu-1}, & \nu<n,\\[2pt]
    0,                                  & \nu=n.
  \end{cases}
\end{equation}
Here the right-hand side is a degree-$n$ intersection number.  The
functional $\ell$ is nonnegative on every effective divisor, because
the other factors in the product are nef (their curvature forms are
semipositive).  It vanishes on $L$ itself when $\nu<n$, because
$\alpha^{\nu+1}=0$ kills the highest power; when $\nu=n$ it is zero
by definition.  So $\ell(L)=0$.  This is exactly the
\emph{curvature-null face}: the collection of divisors whose
intersection with the semipositive curvature vanishes entirely.

The first main result is the following.

\begin{theorem}[Exact attainment of zero order]\label{thm:exact}
Let $X$ be a smooth connected projective complex variety, $L=-K_X$
with a smoothly semipositive Hermitian metric $\alpha$ of maximum
rank $\nu$, $G_0$ ample, and $\ell$ as in \eqref{eq:ell}.  Let $v$ be
either the zero valuation on $\C(X)$ or a discrete valuation centered
at some point of $X$.  Let $M$ be any fixed line bundle.

Suppose that effective integral divisors $N_j$ and positive
integers $m_j\to\infty$ satisfy
\[
  N_j \sim m_j L + M,\qquad
  \ell(N_j)=0,\qquad
  \frac{v(N_j)}{m_j}\longrightarrow 0.
\]
Then there is an effective rational divisor $N\sim_{\Q}L$ with
$v(N)=0$.
\end{theorem}

\begin{remark*}[In words]
Read $\ell(N_j)=0$ as saying: the divisors $N_j$, in their class,
drag their weight as far as possible towards the curvature-null
direction of $L$.  Read $v(N_j)/m_j\to 0$ as saying: they also vanish
with vanishing \emph{density} along the prescribed center.  The
theorem says these two errands can be completed at the same time:
some genuine rational prototype of the anticanonical class
\emph{does not vanish at all} at that center.  The auxiliary bundle
$M$ is arbitrary; it is carried along for free.
\end{remark*}

The second main result turns this analytic statement into an
algebraic one in the presence of a torus action.

\begin{theorem}[Invariant anticanonical nonvanishing]\label{thm:invariant}
Let $X$ be a smooth connected projective complex variety with
$-K_X$ smoothly semipositive and $H^i(X,\OO_X)=0$ for every $i>0$.
For every algebraic torus action $T\curvearrowright X$, some positive
anticanonical multiple $H^0(X,-mK_X)$ contains a nonzero $T$-invariant
section, using the natural linearization on $-K_X$.
\end{theorem}

The natural linearization is the one induced by the determinant of
the action on the tangent bundle; see \S\ref{sec:conventions}.

\begin{remark*}[In words]
A group acting on $X$ can permute sections of $mL$
\emph{projectively}: each group element sends a section to a scalar
multiple of another section.  The theorem says that if the group is a
torus (a product of copies of $\mathbb C^*$) and $X$ has no
unexpected cohomology in the structure sheaf, then some section is
actually fixed, not just scaled, after we pass to a possibly higher
power.  Fixing one section is the algebraic version of the analytic
"don't vanish at a center" statement.
\end{remark*}

The two constructions dovetail into a scalar statement.

\begin{theorem}[$L$-scalar nonvanishing, informal]\label{thm:scalar}
For every smooth connected projective complex variety $X$ with
smoothly semipositive $-K_X$, some $H^0(X,-mK_X)$ with $m>0$ is
nonzero.
\end{theorem}

Theorem~\ref{thm:scalar} is not proved here from nothing: it uses
Theorem~\ref{thm:invariant} together with two structural descents
(Sections~\ref{sec:desc-isometric}--\ref{sec:desc-mrc}), which rely on
companion preprints H and I for the fact that the residual monodromy
of the relevant fibrations is commutative and can be reduced to a
torus.  Restated: the companions give the structure, we give the
fiberwise section, and the descent glues them.

\subsection{Strategy in one page}

Here is the whole proof, compressed into a single page.  If you
understand the following five sentences, you understand the paper.

\begin{enumerate}[leftmargin=2em]
\item \textbf{Null-polar field.}  Consider the rational functions
      $h$ on $X$ whose pole divisor is $\ell$-null.  They form a
      field $k$, relatively algebraically closed in $\C(X)$.  Model
      $k$ by a smooth base $Y$, pick a normal equidimensional
      $W\to Y$ dividing the graph of a rational map $X
      \dashrightarrow Y$, and study the "correction divisor"
      $A_0$ on $Y$ whose pullback is exceptional over $X$.
\item \textbf{Endpoint lemma.}  A finitely generated bigraded ring
      $R\subset K[x,y]$ has generators on the boundary ray $x=0$.
      One such generator witnesses the limiting valuation, without
      needing any information about coefficients we have not seen.
      This reduces a "sum of monomials" problem to a single
      generator.
\item \textbf{Exact attainment.}  Apply the endpoint lemma to the
      two-dimensional section ring of the sequence $N_j/m_j$.  The
      correction $A_0$ is removed by birational pushforward after
      selecting the boundary generator.  The centeredness of $v$
      then converts the resulting upper bound into exact equality.
\item \textbf{Invariant main.}  For a torus action, the source
      component of a one-parameter subgroup attracts a dense open
      set; Atiyah--Bott / Atiyah--Segal / Atiyah--Singer give
      cohomology classes of extremal weight there; hard Lefschetz
      turns them into twisted differential forms; saturated
      determinants produce effective divisors of bounded order
      along the generic orbit; exact attainment produces actual
      section weights whose convex hull contains zero; the product
      is the invariant section.
\item \textbf{Descents.}  Two complementary structural results
      reduce any $X$ to one on which the invariant main theorem
      applies: the compact-monodromy cover (via the
      universal-cover product description) and the locally constant
      MRC fibration (via Matsumura--Wang and Muller's
      reduction).  Finite-\'etale norms turn the fiberwise section
      into one on $X$.
\end{enumerate}

Everything else in the paper is either intuition for these five
sentences or a technical step needed by them.

\subsection{Organization and reading guide}

\begin{itemize}[leftmargin=2em]
\item Section~\ref{sec:conventions} fixes every definition and
      convention we use.
\item Sections~\ref{sec:endpoint}--\ref{sec:exactproof} prove the
      endpoint lemma and then Theorem~\ref{thm:exact}.  Read them
      in one sitting.  Section~\ref{sec:nullfield} constructs the
      null-polar base; Section~\ref{sec:exactproof} does the final
      two-dimensional section-ring step.
\item Section~\ref{sec:arc} derives the formal-arc corollary.
\item Sections~\ref{sec:orderweight}--\ref{sec:exactweights} prove
      Theorem~\ref{thm:invariant}: order-and-character bookkeeping,
      then the source-weight and cohomology argument.
\item Section~\ref{sec:isharedtools} collects the shared tools
      (equidimensional models, bounded maximum metrics, nef descent,
      curvature concentration).  If you are here for the proof of
      Theorem~\ref{thm:exact}, you will only need the metric lemma.
\item Sections~\ref{sec:approxarc}--\ref{sec:approxweights} give an
      alternative "approximate" proof of the same content, which is
      sometimes used in the literature as a stand-alone route.
      It is independent of exact attainment.
\item Section~\ref{sec:invfield} and Section~\ref{sec:ccone} give two
      further invariant-section constructions, neither of which
      requires the first-jet step.  Both use companion preprints.
\item Sections~\ref{sec:desc-isometric} and \ref{sec:desc-mrc} give
      the descents leading to Theorem~\ref{thm:scalar}.
\item The appendix collects the reference list.
\end{itemize}
\section{Conventions and basic objects}
\label{sec:conventions}

Everything below is standard.  We collect it in one place because the
paper uses all of it, and because a reader who has not met a term
should be able to look it up here without paging through references.

\subsection{Complex projective varieties}

A \emph{complex projective variety} is a closed subvariety of some
$\PP^N$ over $\C$.  It is \emph{smooth} if every point has a smooth
neighbourhood (concretely, if the Jacobian of the defining equations
has the expected rank everywhere), \emph{connected} if it has one
connected component, and \emph{integral} if it is connected and the
algebra of functions on any open affine piece has no nilpotents and no
zero divisors.  We will only work with smooth connected projective
varieties, which we abbreviate \emph{scvp} when we need to.
For such an $X$,
\[
  H^0(X,\OO_X) = \C
\]
because a projective variety has only constant holomorphic functions.

An \emph{open subset} is a Zariski-open subset; a \emph{general point}
of $X$ is a point avoiding any union of finitely many proper closed
subsets.  A \emph{very general point} avoids any countable union of
proper closed subsets.  A \emph{divisor} is an irreducible codimension
one subvariety.

\subsection{Line bundles and sections}

A \emph{line bundle} $\mathcal L$ on $X$ is a locally free sheaf of
rank one.  For any open set $U$, a \emph{section} over $U$ is an
element of $H^0(U,\mathcal L)$; equivalently, a choice of trivialisation
times a regular function.  We write $|\mathcal L|$ for the associated
linear system $H^0(X,\mathcal L)$ up to scaling; a basepoint-free
linear system gives a morphism to projective space.

Line bundles form a group under tensor product, denoted $\Pic(X)$.
For $m\in\Z$, $\mathcal L^m$ is the $m$-th tensor power (with
$\mathcal L^{-1}$ the dual).  A line bundle is \emph{ample} if
$\mathcal L^m$ is basepoint-free and gives an embedding for all $m$
large; it is \emph{very ample} if $\mathcal L$ itself gives an
embedding.  Ample implies nef (definition below).

\subsection{Canonical and anticanonical bundles}

Let $n = \dim X$.  The \emph{canonical bundle} is
\[
  K_X \;=\; \wedge^{n}\,\Omega^1_X ,
\]
the top exterior power of the cotangent bundle.  Its dual
\[
  L \;=\; -K_X \;=\; K_X^{-1} \;=\; \wedge^{n}\,T_X
\]
is the \emph{anticanonical bundle}.  A section of $mL$ is a global
holomorphic section of $(K_X^{-1})^{\otimes m}$.  Existence of such a
section is our main question.

If a torus $T$ acts algebraically on $X$, the action differentiates
to an action on $T_X$, hence on $\wedge^n T_X = L$.  This is the
\emph{natural linearization} of $L=-K_X$.  Restated: once the group
acts on $X$, it automatically acts on the anticanonical line, by the
determinant of the differential.

\subsection{Hermitian metrics, curvature, positivity}

A \emph{smooth Hermitian metric} on a line bundle $\mathcal L$ assigns
to each point $x$ a Hermitian norm $|\cdot|_x$ on the fiber $\mathcal
L|_x$, varying smoothly.  In a local frame where a section has value
$f$, the \emph{curvature form} is
\[
  \alpha \;=\; -\frac{i}{2\pi}\,\partial\bar\partial \log|f|^2,
\]
which is independent of the frame.  The curvature form represents the
first Chern class $c_1(\mathcal L)\in H^2(X,\R)$, an integral class.

A form $\alpha$ is \emph{semipositive} if it is positive semidefinite
at every point; \emph{positive} if it is positive definite.  The
\emph{maximum rank} $\nu$ is the largest integer such that the pointwise
rank of $\alpha$ equals $\nu$ on some nonempty open set; for semipositive
$\alpha$ this achieves at a very general point.  We use the phrase
"smoothly semipositive" to mean there exists a smooth Hermitian metric
on $\mathcal L$ whose curvature is semipositive; the metric is not
required to be positive.

Let $D$ be a Cartier divisor with line bundle $\OO_X(D)$.

\begin{itemize}[leftmargin=2em]
\item $D$ is \emph{nef} (numerically effective) if $D\cdot C\ge 0$
      for every irreducible curve $C\subset X$.
\item $D$ is \emph{ample} if a positive multiple is very ample.
\item $D$ is \emph{big} if $D^n>0$ (equivalently, if $h^0(X,mD)$ grows
      like $m^n$).
\item $D$ is \emph{pseudoeffective} if its class is a limit of classes
      of effective divisors.
\item $D$ is \emph{effective} if it is a nonnegative integer
      combination of prime divisors.
\end{itemize}

A line bundle with a semipositive Hermitian metric is nef: the integral
of the curvature over the parametrization of a curve is nonnegative.
The converse is not generally true.  We work with the analytic
hypothesis (a smoothly semipositive metric) because the cohomological
hard-Lefschetz statements we use require it.

\subsection{Divisors, intersection, linear equivalence}

A \emph{$\Q$-Cartier divisor} on a projective variety is a finite sum
$\sum a_i D_i$ with $D_i$ irreducible divisors and $a_i\in\Q$; a
\emph{rational Cartier divisor} means the same.  We say $D_1\sim D_2$
if their difference is the divisor of a rational function (linear
equivalence), and $D_1\sim_{\Q} D_2$ if a positive integer multiple
of their difference is.  Two rational equivalence classes have the
same intersection number with any other class, so they have the same
$\ell$-value in \eqref{eq:ell}.

The notation $D\cdot L^\nu\cdot G_0^{\,n-\nu-1}$ means the
intersection product of divisor classes on the smooth projective $X$:
at each point the product of divisor classes takes values in $\Z$ (for
integral classes) or $\Q$ (for rational classes).  The form
$\alpha^{\nu+1}\equiv 0$ says the intersection product of $(\nu+1)$
copies of the nef class $L$ vanishes, consistent with its having
$\ell(L)=0$.

\subsection{Birational maps, resolution, exceptional divisors}

A \emph{rational map} $f\colon X\dashrightarrow Y$ is an equivalence
class of morphisms from a dense open subset of $X$ to $Y$.  A
\emph{birational map} is a rational map with a rational inverse.  A
\emph{birational morphism} is a morphism that is also birational.

An \emph{exceptional divisor} of a birational morphism $\pi\colon
Z\to X$ is an irreducible prime $E\subset Z$ such that $\dim\pi(E) <
\dim E$; i.e.\ $\pi$ collapses $E$.  The \emph{resolution of
singularities theorem} (Hironaka) says every variety has a birational
projective model that is smooth; we use this freely.

The \emph{Jacobian divisor} of a birational morphism $p\colon W\to X$
of smooth varieties, if it exists, is
\[
  K_{W/X} \;=\; K_W - p^* K_X,
\]
an effective divisor supported on the exceptional locus.  Its natural
section $s_{K_{W/X}}$ is the determinant of the Jacobian matrix of $p$.

\emph{Normalization} of a variety is the finite birational morphism
from its integral closure in its function field.  If $f\colon W\to Y$
has all fibers of the same dimension $d=\dim W-\dim Y$, we say $f$ is
\emph{equidimensional}; this is the natural substitute for "flat"
when both source and target are varieties.  A normal equidimensional
model $W\to Y$ has the property that every vertical prime divisor of
$W$ (one not dominating $Y$) maps onto a prime divisor of $Y$.

\subsection{Torus actions and linearizations}

An \emph{algebraic torus} is an algebraic group $T\cong
(\C^*)^d$.  A \emph{faithful torus action} on $X$ is an action with
trivial kernel on automorphisms of $X$.

A \emph{linearization} of a line bundle $\mathcal L$ with respect to
$T$ is a lift of the action to $\mathcal L$ commuting with the
projection $\mathcal L\to X$.  If a linearization exists, each
$t\in T$ acts on the space of sections $H^0(X,\mathcal L)$ by a trace
map; more precisely, for any section $s$, $t^*s$ is again a section,
and when $X$ is proper this forms a representation of $T$ on
$H^0(X,\mathcal L)$.

The group $T$ splits into characters $\chi\colon T\to\C^*$, a lattice
$X^*(T)\cong\Z^d$.  A section $s$ is an \emph{eigensection} of
character $\chi$ if $t^*s = \chi(t)\,s$ for all $t$.  When
$H^0(X,mL)\ne 0$, the space decomposes into eigenspaces for $T$, one
per character occurring.  The \emph{character} of a section is that
eigencharacter.

\subsection{Valuations, order, and formal arcs}

A \emph{discrete valuation} of the function field $\C(X)$ is a
surjective map $v\colon\C(X)^\times\to\Z$ satisfying $v(fg)=v(f)+v(g)$
and $v(f+g)\ge\min\{v(f),v(g)\}$ when $f+g\ne 0$.  It extends to a
function on Cartier divisors: if $D$ is a Cartier divisor with local
equation $f$ near a point $\xi$, then $v(D)=v(f)$ is well defined
because units have valuation zero.  We also allow the \emph{zero
valuation} $v\equiv 0$, which is consistent with the definition but
has image $\{0\}$ instead of $\Z$.

A valuation is \emph{centered at $\xi\in X$} if its valuation ring
dominates $\OO_{X,\xi}$; equivalently, the maximal ideal of $\OO_{X,\xi}$
is inside the maximal ideal of the valuation ring.  This is the
right comparison: $v$ "lives above" $\xi$.

An \emph{order relation} associated to a valuation restricts to a
function on effective divisors, and on rational divisors by linearity.
For an effective divisor $D$ and a centered valuation $v$, $v(D)$ is
a nonnegative integer.  The zero valuation gives $v(D)=0$ always.

A \emph{formal arc} on $X$ is a morphism
\[
  \gamma\colon\Spec F[[t]]\longrightarrow X
\]
over an extension field $F/\C$.  We will always take $F$ big enough
to be algebraically closed in the relevant sense (our constructions
are algebraic over $\C$, so $F=\C$ suffices unless a companion
requires more).  The \emph{generic formal arc} has the additional
property that the fraction point maps to the generic point of $X$.
For such a $\gamma$ and Cartier divisor $D$, the \emph{order along
$\gamma$} is the order in $t$ of a local equation for $D$ pulled back
along $\gamma$; this is a degree in $t$ and coincides with a
discrete valuation centered at the image of the generic point.  All
formal arcs used below carry a specific center; if that center is
the generic point we use the zero valuation for simplicity.

\subsection{Bia\l ynicki-Birula decomposition and the source component}

Let a one-parameter subgroup $\lambda\colon\C^*\to T$ act on a smooth
projective $X$.  Since $X$ is proper, the limit
\[
  \lim_{t\to 0}\,\lambda(t)\cdot x
\]
exists for every $x$ such that the orbit does not escape to infinity;
in a projective variety the limit always exists.  The set of $x$ for
which the limit lies in a fixed connected component $S$ of the
$\lambda$-fixed locus is a locally closed subvariety of $X$.  The
\emph{source component} $S_+$ is the unique such component whose
attracting set is dense; it exists and is unique by the
Bia\l ynicki-Birula decomposition \cite{BB1973}.  Every other fixed
component attracts a set of smaller dimension, and the source
component dominates because the generic orbit's limit lands in it.

If $\lambda$ is the zero cocharacter or $T$ is trivial, we declare
$S_+=X$ and the fiber weight at $S_+$ to be zero.

\subsection{Multiplier ideals and singular metrics}

Let $\varphi$ be a locally integrable function on an open set $U$ of
a complex manifold.  The assignment $h = e^{-\varphi}$ is a
\emph{singular Hermitian metric}.  Its curvature is the distribution
$\Theta_h = i\partial\bar\partial\varphi$.  The \emph{multiplier
ideal} sheaf $\mathcal I(h)$ or $\mathcal I(\varphi)$ associates to an
open set $V\subset U$ the holomorphic functions $f$ on $V$ such that
$|f|^2 e^{-\varphi}$ is locally integrable on $V$.

A quasi-plurisubharmonic function has \emph{neat analytic
singularities} if locally it is of the form
$\varphi = c\log\sum_j|f_j|^2 + u$ with $c>0$, the $f_j$ holomorphic,
and $u$ smooth.  Such functions arise from resolution of ideals.

The \emph{extension theorem of Cao--Demailly--Matsumura}
\cite[Theorem~1.1]{CDM2017} is our main tool: it produces surjections
on cohomology of the form
\[
  H^0(S,\mathcal O_S(K_S+J)\otimes\mathcal I(h))
  \twoheadrightarrow
  H^0(S,\mathcal O_S(K_S+J)\otimes\mathcal I(h)/\mathcal I(he^{-\Psi})),
\]
under suitable positivity hypotheses on $h$ and neat analytic
singularities on $\Psi$.  We will state each use in context.

\subsection{Nef reduction}

The \emph{nef reduction theorem} of Bauer, Campana, Eckl, Kebekus,
Peternell, Rams, Szemberg and Wotzlaw \cite[Theorem~2.1]{BCEKPRSW2002}:
if $B$ is a nef Cartier divisor on a smooth projective variety $Y$,
there is a dominant almost holomorphic map $Y\dashrightarrow Z$ with
connected general fibers, such that $B\cdot C=0$ on every curve $C$
contracted by the map, and $B\cdot C>0$ on every curve $C$ through
a very general point that is not contracted.  In particular, nef
divisors are trivial on their null directions, and positive on all
other directions.  This turns a nef divisor into a pullback of a
nef divisor from a smaller base, plus corrections over exceptional
loci.

\subsection{Section rings and finite generation}

Let $B_0,B_1$ be Cartier divisors on an integral projective variety
$Y$.  The \emph{bigraded section ring} is
\[
  R(Y;B_1,B_0) \;=\;
  \bigoplus_{a,b\ge 0} H^0(Y,\OO_Y(aB_1+bB_0))\,x^a y^b
  \;\subset\; \C(Y)[x,y].
\]
A standard consequence of the adjoint-ring theorem of Birkar--Cascini--Hacon--McKernan
\cite{BCHM2010} and Cascini--Lazi\'c \cite{CasciniLazic2012} is that
such rings are finitely generated when $B_1,B_0$ are chosen from the
output of the companion correction-ring theorem
(Companion E, Theorem (Finite generation near a correction divisor)).
We will use this in \S\ref{sec:exactproof} and
\S\ref{sec:ccone}.

\subsection{Companion preprints}

Four companion preprints supply theorems we treat as inputs:

\begin{itemize}[leftmargin=2em]
\item \textbf{Companion G}: \emph{Metric descent and rank-preserving
      contractions}.  Supplies the toroidal-volume theorem.
\item \textbf{Companion E}: \emph{Integrable metrics and effectivity
      with controlled boundary}.  Supplies the finiteness and
      weighted-approximation statements for correction divisors.
\item \textbf{Companion H}: \emph{Anticanonical nonvanishing from
      smooth semipositivity}.  Supplies the finite cover with
      compact torus monodromy.
\item \textbf{Companion I}: \emph{Invariant anticanonical indices and
      conversion of twisted differentials}.  Supplies the invariant
      index theorem and its conversion to twisted forms.
\end{itemize}

We will cite each theorem by number exactly when used, but we never
restate the internal construction of a companion paper.  A reader who
wants full closure must chase those references.


\section{The endpoint lemma and its birational application}
\label{sec:endpoint}

This section proves two things.  First, a purely algebraic lemma: in a
finitely generated bigraded ring, a sequence of elements whose
normalized degree approaches the boundary ray $x=0$ must be
"explained" by a single generator that sits on that ray.  Second, a
geometric proposition: given any configuration of divisors, sections,
and valuations that is compatible with finite generation of an
associated section ring, one can produce an exact-order representative
of $D_0$.  These two statements together are the entire kernel of the
proof of Theorem~\ref{thm:exact}.  Everything else in that proof is
getting into the situation where these statements apply.

\subsection{The endpoint lemma}

Let $K$ be a field containing $\C$, and let $v\colon K^\times\to\R$ be
a real-valued valuation trivial on $\C^\times$ (so $v(\C^\times)=0$).
The zero valuation $v\equiv 0$ is allowed.  For a homogeneous
polynomial $f x^a y^b$ with $f\in K$, the valuation is applied to
$f$; the formal variables $x,y$ only record the two degrees.

The \emph{boundary ray} of the bidegree lattice $\N^2$ is the set of
degrees $(0,b)$ with $b>0$.  A sequence $(a_j,b_j)$ with
$a_j,b_j\ge 0$ and $d_j = a_j + b_j > 0$ \emph{approaches the
boundary ray} if $a_j / d_j\to 0$; equivalently $b_j / d_j\to 1$.

\begin{lemma}[Endpoint selection]\label{lem:endpoint}
Let $R\subset K[x,y]$ be a finitely generated $\C$-subalgebra, graded
by $\N^2$ with $\deg x = (1,0)$ and $\deg y = (0,1)$, and with
$R_{0,0}=\C$.  Let $v$ be a real-valued valuation of $K$ trivial on
$\C^\times$.  Suppose that nonzero homogeneous elements
$s_j = f_j x^{a_j} y^{b_j}\in R$ with $d_j = a_j + b_j > 0$ satisfy
\[
  \frac{a_j}{d_j}\longrightarrow 0,\qquad
  \frac{v(f_j)}{d_j}\longrightarrow \gamma\in\R.
\]
Then every finite set of nonzero homogeneous generators of $R$ contains
a generator $f x^0 y^b$ with $b>0$ and
\[
  \frac{v(f)}{b}\le \gamma.
\]
\end{lemma}

\begin{proof}[In words]
Fix any finite homogeneous generating set.  Each $s_j$ is a sum of
monomials in the generators.  Pick one monomial of that degree whose
coefficient has valuation no larger than the coefficient of $s_j$
itself; the triangle inequality guarantees such a monomial exists.
The monomial has total degree $(a_j,b_j)$.  If we assume the generators
with nonzero $x$-part absorb at most $a_j$ total degree in $x$, the
$y$-degree forced into the $x$-free generators must be nearly $b_j$;
the valuation of each such generator scales linearly with its
$y$-exponent.  The minimum ratio $\lambda$ of valuation to $y$-degree
among the $x$-free generators therefore bounds $\gamma$ from above.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:endpoint}]
Discard constant generators.  Write the remaining ones as
\[
  r_i = g_i\,x^{u_i} y^{w_i},\qquad
  u_i,w_i\in\N,\qquad u_i+w_i>0,
  \qquad i = 1,\dots,N.
\]
Since the generators are homogeneous, each $s_j$ is a sum of finitely
many monomials of bidegree $(a_j,b_j)$ in the $r_i$.  Because
$v(f_j)=v(s_j)$ is the valuation of the coefficient of $s_j$, and
the valuation of a sum is at least the minimum of the valuations of
its summands, at least one monomial has coefficient valuation at most
$v(f_j)$; otherwise the sum would have valuation larger than $v(f_j)$.
Let $e_{ij}$ be the exponents of the generators in such a monomial,
and let $c_{ij}\in\C^\times$ be its coefficient.  Since $v$ is
trivial on $\C^\times$, the nonconstant scalar coefficients contribute
nothing, so
\begin{equation}\label{eq:monomial}
  \sum_i e_{ij} u_i = a_j,\qquad
  \sum_i e_{ij} w_i = b_j,\qquad
  \sum_i e_{ij}\,v(g_i)\le v(f_j).
\end{equation}

Set
\[
  C_1 \;=\; \max_i w_i,\qquad
  C_2 \;=\; \max_i \max\{0,-v(g_i)\}.
\]
Both are nonnegative constants, depending only on the chosen generating
set.  Split the first equation of \eqref{eq:monomial} by whether
$u_i>0$ or $u_i=0$.  Each $u_i>0$ contributes at least $1$ to the
left side, so
\[
  \sum_{u_i>0} e_{ij} \;\le\; a_j.
\]
Consequently
\[
  \sum_{u_i>0} e_{ij}\,w_i \;\le\; C_1\,a_j,
  \qquad
  \sum_{u_i>0} e_{ij}\,v(g_i) \;\ge\; -C_2\,a_j.
\]
Some generator must have $u_i = 0$: otherwise the first equation of
\eqref{eq:monomial} would give $b_j \le C_1 a_j$, contradicting
$a_j/d_j\to 0$ since $d_j = a_j + b_j$ gives $b_j/d_j\to 1$.  Every
$x$-free generator has $w_i>0$ because constant generators were
discarded and $R_{0,0}=\C$.  Define
\[
  \lambda \;=\; \min_{u_i=0}\,\frac{v(g_i)}{w_i}.
\]
The minimum is finite because the set is nonempty.

Let $c_j = \sum_{u_i>0} e_{ij} w_i$, so $0\le c_j \le C_1 a_j$.  The
$x$-free generators contribute $b_j - c_j$ to the $y$-degree, and
their combined valuation is at least $\lambda(b_j - c_j)$ by
definition of $\lambda$.  The $x$-bearing factors contribute at least
$-C_2 a_j$ to the valuation.  Equation \eqref{eq:monomial} therefore
implies
\[
  v(f_j) \;\ge\; \lambda(b_j - c_j) - C_2 a_j
          \;\ge\; \lambda b_j - (C_2 + |\lambda|C_1)\,a_j .
\]
Divide by $d_j = a_j + b_j > 0$ and use $a_j/d_j\to 0$ and
$b_j/d_j\to 1$:
\[
  \gamma \;=\; \lim_j \frac{v(f_j)}{d_j}
         \;\ge\; \lambda .
\]
The finite minimum $\lambda$ is realized by some generator with
$u_i = 0$ and $w_i > 0$, so its ratio $v(g_i)/w_i = \lambda \le
\gamma$.  This proves the lemma.
\end{proof}

The proof uses the actual coefficient functions $g_i$ of the chosen
finite generating set: knowing only an abstract limiting cone of
numerical classes would not suffice.  This is precisely the feature
that will make the geometric application below work.

\subsection{Removing a birational correction}
\label{sec:geometric}

We now state and prove the geometric proposition that converts a
sequence of effective divisors, all of whose normalized orders vanish
along a fixed center, into an \emph{exact} effective representative of
the limiting class.  The key geometric feature is a rational map to a
base $Y$; the correction $A$ on $Y$ has positive valuation in general,
but it is annihilated by birational pushforward, so its value never
needs to be computed.

Relatively few technical hypotheses are needed.  The section ring must
be finitely generated, and the minimum divisor $P(t)$ must be affine in
$t$ on the relevant segment.

\paragraph{Section rings.}
Let $Y$ be an integral projective variety over $\C$ and $B_1,B_0$
Cartier divisors on $Y$.  Identify global sections of $\OO_Y(B)$ with
rational functions bounded by $B$:
\[
  H^0(Y,\OO_Y(B))
  \;=\;\{h\in\C(Y)^\times : \divisor_Y(h)+B\ge 0\}\cup\{0\}.
\]
This identification is standard and gives $H^0(Y,\OO_Y) = \C$ when
$Y$ is projective and integral.  The \emph{bigraded section ring} is
\begin{equation}\label{eq:section-ring}
  R(Y;B_1,B_0) \;=\;
  \bigoplus_{a,b\in\N} H^0(Y,\OO_Y(aB_1+bB_0))\,x^a y^b
  \;\subset\; \C(Y)[x,y],
\end{equation}
graded by $\deg x = (1,0)$, $\deg y = (0,1)$, with $R_{0,0}=\C$.

\paragraph{Rational powers.}
For a rational function $h$ and a rational number $r\in\Q_{>0}$, we use
the formal symbol $h^r$ inside the multiplicative group
$\C(Y)^\times\otimes_\Z\Q$.  Every element of this group has a
positive integer power that is an ordinary rational function.  The
divisor map and every real-valued valuation extend uniquely $\Q$-linear.
No extension of the function field by actual roots is intended.

\paragraph{Centered valuations.}
A real-valued valuation $v$ of $\C(X)$ is \emph{centered on $X$} if
its valuation ring dominates $\OO_{X,\xi}$ for some point $\xi\in X$.
For a Cartier divisor $D$, $v(D)$ means the valuation of a local
equation at that center, and is independent of the choice of equation.
Every effective rational Cartier divisor has $v(D)\ge 0$; the zero
valuation corresponds to a center at the generic point.

\begin{proposition}[Exact effectivity from vanishing normalized order]
\label{prop:geometric}
Let $X,Y$ be smooth integral projective complex varieties, let $W$ be
normal and projective, and let
\[
  X \;\xleftarrow{\;p\;} \; W \;\xrightarrow{\;f\;} \; Y
\]
be morphisms with $p$ birational, and $f$ surjective and
equidimensional.  Identify $\C(Y)$ with its image in
$\C(X)=\C(W)$.  Fix a centered discrete valuation $v$ of $\C(X)$,
allowing the zero valuation.

Let $A\ge 0$ be a rational divisor on $Y$ such that $p_*f^*A = 0$.
Let $D_0,D_1$ be rational Cartier divisors on $X$.  For
$0\le t\le 1$ put $D(t) = (1-t)D_0 + tD_1$.  For every prime divisor
$Q$ of $Y$, define
\begin{equation}\label{eq:minimum-divisor}
  P(t) \;=\; \sum_Q
    \min_{E\,:\,f(E)=Q}
    \frac{\coeff_E p^*D(t)}{\ord_E (f^*Q)}\,Q,
\end{equation}
where the minimum is over prime divisors $E$ of $W$ that dominate $Q$.
Suppose $P(t) = (1-t)P(0) + tP(1)$ throughout $[0,1]$.

Suppose that rational numbers $t_j\in[0,1]$ tend to $0$ and that
$h_j\in\C(Y)^\times\otimes_\Z\Q$ satisfy
\[
  N_j \;=\; D(t_j)+\divisor_X(h_j) \;\ge\; 0,
  \qquad v(N_j) \longrightarrow 0.
\]
Finally suppose that, for some $\epsilon\in\Q_{>0}$ and $k>0$, the
divisors
\[
  B_1 \;=\; k\bigl(A+\epsilon P(1)\bigr),\qquad
  B_0 \;=\; k\bigl(A+\epsilon P(0)\bigr)
\]
are Cartier, and the section ring $R(Y;B_1,B_0)$ is finitely generated.

Then there is an effective rational divisor
\[
  N \sim_{\Q} D_0 \qquad\text{with}\qquad v(N)=0.
\]
\end{proposition}

Several remarks before the proof.

The minimum in \eqref{eq:minimum-divisor} is over a nonempty finite set
because $f$ is surjective and equidimensional.  Only finitely many base
primes occur because $p^*D_0$ and $p^*D_1$ have finite fixed support.
Each coefficient of $P(t)$ is a minimum of finitely many rational
affine functions of $t$, so $P(t)$ is piecewise affine.  In
applications one restricts to a sufficiently small rational segment
ending at $t=0$ and rescales the parameter so the affine hypothesis
holds.  If $Y$ is a point, the sum over $Q$ is empty, $P(t)\equiv 0$,
and the argument below still applies with $A=0$.

\begin{proof}[Proof of Proposition~\ref{prop:geometric}]
Put $t_j = a_j / d_j$ with $d_j = a_j + b_j > 0$, $a_j,b_j\in\N$.
Multiplying both by a positive integer clears denominators without
changing $t_j$, so we may assume
\[
  s_j \;=\; h_j^{\epsilon k d_j}
\]
is an ordinary rational function on $Y$.

\emph{Effectivity on the base.}  The divisor $N_j$ is effective and
rational Cartier on the smooth $X$, so $p^*N_j\ge 0$.  Let $E$ be a
prime divisor of $W$ dominating a base prime $Q$, and put
$e_E = \ord_E(f^*Q) > 0$.  Taking the coefficient of $E$ in
\[
  p^*N_j \;=\; p^*D(t_j) + f^*\divisor_Y(h_j)
\]
gives
\[
  \frac{\coeff_E p^*D(t_j)}{e_E} + \coeff_Q \divisor_Y(h_j)\ge 0.
\]
The minimum over $E$ therefore gives
\begin{equation}\label{eq:base-effectivity}
  P(t_j)+\divisor_Y(h_j)\ge 0.
\end{equation}

\emph{Producing an element of $R$.}  The affine hypothesis implies
\[
  a_j B_1 + b_j B_0
  \;=\; k d_j \bigl(A + \epsilon P(t_j)\bigr).
\]
Since $A\ge 0$ and \eqref{eq:base-effectivity} holds, the section
$s_j x^{a_j} y^{b_j}$ is a nonzero homogeneous element of
$R(Y;B_1,B_0)$ of bidegree $(a_j,b_j)$.

\emph{Valuation limit.}  The valuation $v(D(t))$ is affine in $t$
because the support of $p^*D(t)$ is contained in the fixed support of
$p^*N_1 + p^*N_2$, and valuations are $\Q$-linear on the group
$\C(Y)^\times\otimes_\Z\Q$.  Hence
\[
  v(h_j) = v(N_j)-v(D(t_j)) \longrightarrow -v(D_0),
  \qquad
  \frac{v(s_j)}{d_j}\longrightarrow -\epsilon k\,v(D_0).
\]

\emph{Applying the endpoint lemma.}  We apply Lemma~\ref{lem:endpoint}
to the section ring $R(Y;B_1,B_0)$, the restriction of $v$ to $\C(Y)$,
and the sequence $s_j x^{a_j} y^{b_j}$.  Obtain a homogeneous generator
of degree $(0,b)$ with $b>0$, represented by a rational function $r$,
satisfying
\[
  \frac{v(r)}{\epsilon k b} \le -v(D_0).
\]
Put $h_* = r^{1/(\epsilon k b)}$ in the rational-power group.  The
section inequality for $r$ gives, after rescaling the exponent,
\begin{equation}\label{eq:endpoint-effectivity}
  P(0) + \frac{A}{\epsilon} + \divisor_Y(h_*)\ge 0.
\end{equation}

\emph{Upstairs effectivity and pushforward.}  Every vertical prime
divisor of $W$ dominates a base prime $Q$, by equidimensionality of
$f$.  For such an $E$ over $Q$, the minimum defining $P(0)$ and
\eqref{eq:endpoint-effectivity} show that the coefficient of $E$ in
\[
  p^*D_0 + f^*\Bigl(\divisor_Y(h_*)+\frac{A}{\epsilon}\Bigr)
\]
is nonnegative.  Every horizontal prime divisor of $W$ (one dominating
$Y$) has coefficient of $p^*D_0$ nonnegative because $p^*N_j\ge 0$ for
all $j$ and $p^*D_0$ is a limit of effective divisors along a finite
support.  Consequently the whole divisor is effective on $W$:
\[
  p^*D_0 + f^*\Bigl(\divisor_Y(h_*)+\frac{A}{\epsilon}\Bigr)\ge 0.
\]
Push this forward by $p$.  The assumption $p_*f^*A=0$ kills the
correction term, while pullback and pushforward of the principal divisor
give $\divisor_X(h_*)$.  We obtain
\[
  N = D_0+\divisor_X(h_*)\ge 0,
  \qquad N\sim_{\Q}D_0.
\]

\emph{Exact order.}  Centering of $v$ on $X$ gives $v(N)\ge 0$.
The generator chosen by the endpoint lemma gives
$v(N) = v(D_0)+v(h_*)\le 0$.  Hence $v(N)=0$.
\end{proof}

The proof never computes the valuation of the correction $f^*A$.  Its
removal is a pure divisor identity under birational pushforward.
This is why the fixed valuation can be centered at a point of any
codimension: the correction lives upstairs on a birational model, and
the birational pushforward forgets it entirely.


\section{The curvature-null field and its model}
\label{sec:nullfield}

This section constructs the geometric stage on which the endpoint
lemma will be applied.  The construction has three layers.

First, we carve out a field $k\subset\C(X)$ by asking that the pole
divisor of a rational function not see the curvature of $-K_X$ at all:
its $\ell$-value is zero.  This field is relatively algebraically
closed in $\C(X)$ and finitely generated, so it has a projective model
$Y$; the rational map $X\dashrightarrow Y$ has a graph up to birational
change.

Second, we build a \emph{normal equidimensional} model $W\to Y$ over
which the birational map $X\dashrightarrow Y$ resolves.  Equidimensionality
is the key point: it says every vertical prime divisor upstairs
dominates a prime divisor downstairs, which is exactly what makes the
minimum in formula~\eqref{eq:minimum-divisor} behave.

Third, we compute two analytic facts (the ``rank one'' statement for
the Jacobian line and a strict curvature patch) which feed into the
toroidal-volume theorem of Companion~G.  That theorem then produces
the correction divisor $A_0$ whose pullback is exceptional over $X$.
A correction-ring theorem from Companion~E upgrades this to a
finitely generated section ring.

\subsection{The null divisor class and the null-polar field}

Throughout this section, the data
$X,L,\alpha,\nu,G_0,\ell$ are as in \S\ref{sec:intro}.  An effective
divisor is \emph{null} if it has $\ell$-value zero.  Because $\ell$ is
nonnegative on every effective divisor and additive, every effective
divisor bounded above by a null divisor is itself null.

\begin{lemma}[Null-polar field]\label{lem:polar-field}
The set
\[
  k \;=\; \{0\}\cup\{h\in\C(X)^\times : \ell((h)_\infty) = 0\}
\]
is a subfield of $\C(X)$, relatively algebraically closed in $\C(X)$,
and finitely generated over $\C$.
\end{lemma}

\begin{proof}[In words]
A field whose poles are all $\ell$-null should be closed under
addition, multiplication and inversion.  Addition and multiplication
add pole divisors, so the first two are easy.  Inversion is fine
because zeros and poles of the same rational function are linearly
equivalent, and $\ell$ is linear-equivalence invariant.  Relative
algebraic closedness is the substantive point: a root of a monic
equation over $k$ can have poles only where the coefficients do.
Finiteness is a general fact about subfields of finitely generated
field extensions.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:polar-field}]
If $h_1,h_2\in k$ then $(h_1+h_2)_\infty\le (h_1)_\infty+(h_2)_\infty$,
and $(h_1h_2)_\infty\le (h_1)_\infty+(h_2)_\infty$; both right-hand
sides are effective and bounded by a null divisor, hence null.  So $k$
is closed under $+$ and $\cdot$.  If $h\ne 0$ is in $k$, then $(h)_0$
and $(h)_\infty$ are linearly equivalent, so they have the same
$\ell$-value; therefore $\ell((h^{-1})_\infty)=\ell((h)_0) = 0$, and
$h^{-1}\in k$.  This proves that $k$ is a subfield.

For relative algebraic closedness, let $b\in\C(X)$ be algebraic over
$k$ with monic equation
$b^r + a_1 b^{r-1} + \cdots + a_r = 0$, $a_i\in k$.  At any prime
divisor $P$ of $X$ where $b$ has a pole, the term $b^r$ has valuation
$r\,\ord_P(b)$, and cannot have strictly smaller valuation than every
other term by the ultrametric inequality.  Thus for some $i$,
\[
  i\,\ord_P(b)\;\ge\; \ord_P(a_i) \;\ge\; -\ord_P((a_i)_\infty).
\]
Summing over the finitely many primes at which any $a_i$ has a pole
gives $(b)_\infty\le \sum_i (a_i)_\infty$.  That right-hand side is
null because $k$ is closed under addition.  So $b\in k$.

For finite generation, choose a transcendence basis
$t_1,\ldots,t_d$ for $k/\C$.  The algebraic closure of
$\C(t_1,\ldots,t_d)$ inside the finitely generated field $\C(X)$ is
a finite extension of $\C(t_1,\ldots,t_d)$.  The intermediate field
$k$ is algebraic over that same finitely generated base, hence is
finite over it, hence is finitely generated.
\end{proof}

\subsection{Models and the null-rank lemma}

Choose smooth projective models and morphisms
\[
  X \;\xleftarrow{\;\pi\;}\; Z \;\xrightarrow{\;g\;}\; Y,\qquad
  \C(Y) = k,
\]
with $\pi$ birational and $g$ surjective.  Since $\C$ has characteristic
zero and $k$ is relatively algebraically closed in $\C(X)$, the
generic fiber of $g$ is geometrically integral.

The following lemma gives two facts that will persist under every
birational change of these models.  Both are used in the
toroidal-volume argument later.

\begin{lemma}\label{lem:null-rank}
For every very ample divisor $H$ on $Y$,
\begin{equation}\label{eq:ample-null}
  \ell(\pi_* g^*H) = 0.
\end{equation}
If $E_Z = K_{Z/X}$ is the effective Jacobian divisor of $\pi$, then
\begin{equation}\label{eq:zero-rank}
  \dim_{\C(Y)} H^0(Z_\eta,\OO_Z(E_Z)|_{Z_\eta}) = 1.
\end{equation}
Here $Z_\eta$ is the generic fiber of $g$.
\end{lemma}

\begin{proof}[In words]
The first statement says the field $k$ was chosen exactly so that
pullbacks of very ample divisors downstairs push forward to
$\ell$-null divisors upstairs.  The second statement says the
Jacobian divisor of the birational map $\pi$ has only one independent
section on the generic fiber: the Jacobian section itself.  Any other
generic section, divided by it, becomes a rational function whose
poles are bounded by a pullback, hence null.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:null-rank}]
Choose a basis $s_0,\ldots,s_N$ of the very ample system $|H|$ with
$s_0\ne 0$.  The functions $s_i/s_0$ belong to $k$ by definition of
$\C(Y)=k$.  Their pole divisors on $X$ are therefore all bounded by a
single null divisor $D$, and their pullbacks to $Z$ by $\pi^*D$.
Basepoint freeness says that the componentwise maximum of the polar
divisors of $g^*s_i$ on $Z$ is $g^*\divisor(s_0)$.  Therefore
\[
  (s_0)_\infty \;\text{on } Z \;\le\; \pi^*D,
\]
and pushing forward to $X$,
\[
  0 \;\le\; \ell(\pi_*g^*H) \;\le\; \ell(D) = 0.
\]
This proves \eqref{eq:ample-null}.

For \eqref{eq:zero-rank}: the Jacobian section $s_{E_Z}$ is one
nonzero section.  Any other generic-fiber section of $\OO_Z(E_Z)$,
divided by $s_{E_Z}$, gives a rational function $a\in\C(Z)=\C(X)$
with horizontal pole divisor bounded by $E_Z$.  There are finitely
many remaining polar primes; each maps into a proper closed subset
of $Y$ because $E_Z$ is exceptional over $X$.  A sufficiently high
very ample system $|hH|$ has an effective member $B$ vanishing to
prescribed orders on those images; increasing the orders ensures
\[
  (a)_\infty\;\le\; E_Z+g^*B.
\]
Push forward by $\pi$: after deleting the exceptional divisor $E_Z$
(which satisfies $\pi_*E_Z=0$), the pole divisor $(a)_\infty$ pushed
to $X$ is at most $\pi_*g^*B$.  By \eqref{eq:ample-null},
$\ell(\pi_*g^*B)=0$.  Thus $\ell((a)_\infty)=0$ and $a\in k$.  Since
the field $\C(Y)$ is $k$ identified inside $\C(X)$, $a$ is a constant
on the generic fiber.  Every generic-fiber section of $\OO_Z(E_Z)$ is
a scalar multiple of $s_{E_Z}$, as claimed.
\end{proof}

\begin{remark}
Equality \eqref{eq:zero-rank} is the full adjoint space in degree
zero, not merely an eventual rank statement.  It identifies the
integrated Jacobian volume with a line metric on the base and is
one of the two analytic inputs of the toroidal-volume theorem below.
\end{remark}

\subsection{An equidimensional model without a field extension}

Assume $d=\dim Y>0$ (otherwise $Y$ is a point and
\eqref{eq:minimum-divisor} is trivial).  We need a normal projective
variety $W$ and a diagram
\begin{equation}\label{eq:model}
  X \;\xleftarrow{\;p\;}\; W \;\xrightarrow{\;f\;}\; Y
\end{equation}
with $p$ birational, $Y$ smooth projective, and $f$ equidimensional
and \emph{toroidal} for strict toroidal embeddings.  All modifications
in this construction are birational; in particular the function field
of the base remains $k$.

\paragraph{Toroidal embeddings.}
A \emph{toroidal embedding} of a smooth variety into a smooth
ambient scheme is a pair (ambient, open dense torus-like part) locally
modeled, on each stratum, by a toric variety and its dense torus.  The
complement of the open part is the \emph{boundary}.  Strictness means
that the irreducible components of the boundary are normal.  A
\emph{toroidal morphism} is a morphism of toroidal embeddings that is
locally a toric morphism; on each stratum the corresponding toric
morphism is determined by a lattice-linear map of rational cones.
This is the Abramovich--Karu--Denef formalism.

\paragraph{Weak toroidalization.}
The toroidalization theorem of Abramovich--Denef--Karu
\cite[Theorem~1.1]{ADK2013} first gives a birational toroidal diagram
with smooth projective target.  For a toroidal morphism,
equidimensionality is equivalent to the condition that every source
cone maps onto a target cone; this is Abramovich--Karu
\cite[Lemma~4.1 and Remark~4.2]{AK2000}.

Refine the target cone complex so that the image of every source cone
is a union of cones.  Projective refinements exist by
\cite[Lemma~4.3]{AK2000}; a further projective toroidal resolution
makes the target nonsingular.  Intersecting source cones with the
inverse images of target cones gives projective source subdivisions
and a compatible toroidal morphism.  In a target cone of the new
subdivision, an old cone image meets it in a union of faces which is
convex, hence in a face.  Thus the new source cones map onto target
cones.  This is the equidimensionalization of
\cite[Proposition~4.4]{AK2000}.

In local toric models, a source stratum maps submersively onto its
target stratum.  Cone surjectivity bounds the dimension of every
fiber stratum above by $n-d$; the opposite bound for full fibers
follows because the target is smooth.  This gives equidimensionality.
The normal source may remain singular, and no covering to reduce
multiplicities is used.  Resolve by $Z\to W$ when using smooth
differential forms; that resolution need not itself be equidimensional.
We continue to write $\pi\colon Z\to X$ and $g\colon Z\to Y$ for
the induced maps.

\subsection{The horizontal curvature patch}
\label{sec:patch}

This subsection verifies the remaining analytic hypothesis of the
toroidal-volume theorem.  Let $\omega_0$ be a positive
$(1,1)$-form representing $c_1(G_0)$, and let $\omega_H$ represent a
very ample class on $Y$.  When $\nu<n$, equation~\eqref{eq:ample-null}
implies
\begin{equation}\label{eq:vanishing-form}
  (\pi^*\alpha)^\nu
  \wedge g^*\omega_H
  \wedge \pi^*\omega_0^{\,n-\nu-1} \;=\; 0
\end{equation}
as a top-degree form.  Indeed the left side is a semipositive
top-degree form whose integral over $Z$ equals the intersection number
$\pi^*L^\nu\cdot g^*H\cdot \pi^*G_0^{n-\nu-1}$, which by the
projection formula and \eqref{eq:ample-null} equals $\ell(\pi_* g^*H)
=0$.  A nonnegative top form with zero integral is zero.

At a point of maximum complex rank of $\pi^*\alpha$, away from the
Jacobian divisor of $\pi$, choose complex coordinates orthonormal for
$\pi^*\omega_0$.  Then $\pi^*\alpha$ is diagonal in these coordinates:
its first $\nu$ eigenvalues are positive and the rest are zero.  The
left side of \eqref{eq:vanishing-form} is a positive multiple of the
sum of the diagonal entries of $g^*\omega_H$ in the remaining
$n-\nu$ directions (in an orthonormal frame of $\pi^*\omega_0$).  Those
entries vanish.  A semipositive Hermitian form that vanishes on a
vector has zero mixed terms with that vector, so
\begin{equation}\label{eq:kernel-inclusion}
  \ker \pi^*\alpha \;\subset\; \ker dg .
\end{equation}
For $\nu=n$, the inclusion \eqref{eq:kernel-inclusion} is automatic
(both kernels are zero).

The maximum-rank locus is nonempty and open, and it intersects the
dense smooth base-change locus of $g$ away from the Jacobian divisor
of $\pi$ (both of these are Zariski-dense opens in a smooth
projective variety).  Near a point in their intersection the rank is
constant.  On a smaller relatively compact neighbourhood, the
positive eigenvalues are bounded below (they reach their
smoothly-varying maximum value $\nu$), and \eqref{eq:kernel-inclusion}
gives
\begin{equation}\label{eq:strict-patch}
  \pi^*\alpha \;\ge\; c\, g^*\omega_H
  \qquad (c>0).
\end{equation}
The Jacobian section is nonzero on this patch.  If $\nu=0$, then
\eqref{eq:kernel-inclusion} would force $dg=0$ on a nonempty open
neighbourhood; since $g$ is dominant this is impossible, so a
positive-dimensional base cannot occur in that case (this is the
content of Theorem~\ref{thm:exact} being trivial when $\nu=0$).

\subsection{The toroidal-volume theorem and the correction divisor}
\label{sec:correction}

We have now proved both analytic inputs from the null-polar field:
the rank-one identity \eqref{eq:zero-rank}, and the strict curvature
patch \eqref{eq:strict-patch}.  We now invoke the third companion
input.

\begin{theorem}[Toroidal volume metric, Companion G; see
Theorem (Toroidal volume metric)]
\label{thm:toroidal-volume}
Under the hypotheses of \eqref{eq:model}, \eqref{eq:zero-rank} and
\eqref{eq:strict-patch}, there is an effective rational divisor
$A_0\ge 0$ on $Y$ satisfying $p_* f^* A_0 = 0$, and a semipositive
metric on $-K_Y+A_0$ whose squared frame norms are locally integrable
and which is smooth and strictly positive on a nonempty ordinary open
subset of $Y$.
\end{theorem}

The theorem's proof, in Companion~G, gives an explicit formula for the
correction:
\begin{equation}\label{eq:correction}
  A_0 \;=\; \sum_Q \max\Bigl\{0,\,
       \min_{E\mapsto Q}\Bigl(\frac{1+a_E}{\ord_E(f^*Q)}-1\Bigr)
    \Bigr\}\,Q,
\end{equation}
where $a_E = \ord_E(K_{W/X})$ is the coefficient of $E$ in the
Jacobian divisor of $p\colon W\to X$, and the sum is over prime
divisors $Q$ of $Y$.  This is an effective rational divisor.  The
identity $p_*f^*A_0 = 0$ is part of the conclusion.

We now apply the correction-ring theorem of Companion~E.

\begin{theorem}[Finite generation near a correction divisor, Companion E;
see Theorem (Finite generation near a correction divisor)]
\label{thm:correction-ring}
Let $A_0\ge 0$ be as above, with the semipositive metric of
Theorem~\ref{thm:toroidal-volume}.  For every finite list of rational
divisors $P_i$ on $Y$, there exist $\epsilon\in\Q_{>0}$ and
$k_0\in\Z_{>0}$ such that all $k_0(A_0+\epsilon P_i)$ are Cartier,
and the multigraded section ring associated with the directions
$(A_0+\epsilon P_i)$ is finitely generated.
\end{theorem}

The section ring in Theorem~\ref{thm:correction-ring} is the
multigraded generalisation of the bigraded ring \eqref{eq:section-ring}.
We will only use the two-variable version stated in
\S\ref{sec:geometric}.

We are ready to prove Theorem~\ref{thm:exact}.  We will do that in
the next section; for now we record that all the geometric data is in
place.

\subsection{Summary of the input to Theorem~\ref{thm:exact}}

Take the two input divisors $N_1,N_2$ from the statement of
Theorem~\ref{thm:exact}.  Set $D_0,D_1$ to be the two linear
combinations
\[
  D(t) \;=\; t^1 N_1 + t^2 N_2,\qquad 0\le t^1,t^2\le 1,
\]
for varying rational $t = (t^1,t^2)$.  In \S\ref{sec:exactproof}
we choose a short segment $t_* \to t_0$ of the ray
$u_1 t^1 + u_2 t^2 = 1$ where $u_j$ is the exponent of $N_j$ (so that
$N_j/m_j=D$ along the ray), and apply Proposition~\ref{prop:geometric}
with the two endpoints of the segment as the two Cartier divisors.

The necessary hypotheses are supplied as follows:
\begin{itemize}[leftmargin=2em]
\item the equidimensional model $W\to Y$ is from
      \S\ref{sec:nullfield};
\item the correction $A_0$ and its metric are from
      Theorem~\ref{thm:toroidal-volume};
\item the finite generation of the section ring $R(Y;B_1,B_0)$ is from
      Theorem~\ref{thm:correction-ring} with the choices
      $P_i = P(t_0),P(t_*)$;
\item the minimum divisor $P(t)$ is affine on the chosen segment
      because it is a minimum of finitely many rational affine
      functions of $t$;
\item the normalized effective divisors $N_j/m_j = D(t_j) + \divisor_X(h_j)$
      approach $D(t_*)$ from the appropriate side along the chosen
      segment, with principal corrections $h_j$ extracted from the
      ambient function field $k$;
\item $v(N_j)/m_j\to 0$ is the hypothesis of
      Theorem~\ref{thm:exact}.
\end{itemize}


\section{Proof of exact attainment}
\label{sec:exactproof}

We are now in a position to prove Theorem~\ref{thm:exact}.  The
argument is short once the prerequisites of
Sections~\ref{sec:endpoint} and \ref{sec:nullfield} are in hand.
We will proceed as follows.

\begin{enumerate}[leftmargin=2em]
\item Pick two indices $N_1,N_2$ from the sequence in the statement,
      with distinct exponents $m_1<m_2$, and organise them into a
      two-parameter family $D(t) = t^1 N_1 + t^2 N_2$ whose values lie
      along a line of anticanonical-class combinations.
\item On this line, the sequence $N_j/m_j$ sits at the points $t_j$.
      These approach a fixed rational point $t_*$ on the line, where
      the class becomes exactly $L$.
\item Write the principal corrections as elements $h_j$ of
      $k^\times\otimes_\Z\Q$.
\item Use the geometric Proposition~\ref{prop:geometric} along a short
      segment from $t_*$ to a choice $t_0$ that stays affine for the
      minimum divisor $P(t)$.  The proposition produces an effective
      rational divisor $N\sim D(t_*)\sim_{\Q}L$ with $v(N)=0$.
\end{enumerate}

The companion results (Sections~\ref{sec:correction}) guarantee all
the hypotheses.

\subsection{The two-parameter family}

Choose two indices with $m_1<m_2$, and let $N_1,N_2$ be the
corresponding divisors with exponents $u_1:=m_1$ and $u_2:=m_2$.
For a rational vector $t=(t^1,t^2)$, put
\[
  D(t) \;=\; t^1 N_1 + t^2 N_2.
\]
Each $D(t)$ is a rational Cartier divisor on $X$ with class
\[
  D(t) \;\sim_{\Q}\; (u_1 t^1 + u_2 t^2)\,L + (t^1 + t^2)\,M .
\]
For each $j$, choose $t_j$ solving
\begin{equation}\label{eq:t-rays}
  u_1 t_j^1 + u_2 t_j^2 = 1, \qquad
  t_j^1 + t_j^2 = 1/m_j .
\end{equation}
Then $D(t_j) \sim_{\Q} L + M/m_j$, the same rational class as
$N_j/m_j$.

Define $t_*$ by the same first equation, but with
$t_*^1 + t_*^2 = 0$:
\[
  u_1 t_*^1 + u_2 t_*^2 = 1, \qquad
  t_*^1 + t_*^2 = 0 .
\]
Explicitly $t_*^1 = 1/(u_1-u_2)$ and $t_*^2 = -1/(u_1-u_2)$, which are
rational because $u_1\ne u_2$.  Consequently $D(t_*) \sim_{\Q} L$:
the $M$-coefficient vanishes.

The sequence $t_j$ approaches $t_*$: from
\eqref{eq:t-rays} we get $t_j^2 = (1-u_1/m_j)/(u_2-u_1)$ and
$t_j^1 = 1/m_j - t_j^2$, and both converge to the limiting values as
$j\to\infty$.

\subsection{Principal corrections lie in the null-polar field}

Because $D(t_j)$ and $N_j/m_j$ have the same rational class, there are
rational functions $h_j$ such that
\begin{equation}\label{eq:principal-corrections}
  N_j/m_j \;=\; D(t_j) + \divisor_X(h_j).
\end{equation}
A positive power of $h_j$ clearing denominators is an ordinary
rational function on $X$.  Its polar divisor is bounded by a positive
linear combination of $N_1$ and $N_2$, because the left side of
\eqref{eq:principal-corrections} is effective.  That bound is null
(Section~\ref{sec:nullfield}, hypothesis $\ell(N_j)=0$).  So the power
belongs to $k$, and we may regard $h_j$ as an element of
$k^\times\otimes_{\Z}\Q$.

\subsection{Case $\dim Y = 0$}

If $k=\C$ (equivalently, $Y$ is a point in the model
\eqref{eq:model}), then every principal correction vanishes:
$h_j\in\C^\times$.  The divisors $D(t_j)$ are effective, and their
support is contained in the fixed support of $p^*N_1+p^*N_2$, so the
limit $D(t_*)$ is an effective rational divisor (as a limit of
effective divisors along a fixed support) with class $L$.  Linearity
of $v$ on the fixed support gives
\[
  v(D(t_*)) \;=\; \lim_j v(N_j)/m_j \;=\; 0.
\]
Then $N=D(t_*)$ proves the theorem in this case.

\subsection{The general case}

Assume $\dim Y>0$.  Use the equidimensional model $W\to Y$ from
\S\ref{sec:nullfield}.  Define the minimum divisor
\begin{equation}\label{eq:P-of-t}
  P(t) \;=\; \sum_Q
    \min_{E:\,f(E)=Q}
    \frac{\coeff_E p^*D(t)}{\ord_E f^*Q}\,Q
\end{equation}
on $Y$, exactly as in \eqref{eq:minimum-divisor}.  Its support is
contained in a fixed finite set (the primes occurring in
$p^*N_1+p^*N_2$).  Each coefficient is a minimum of finitely many
rational affine functions of $t$, so $P(t)$ is piecewise affine in
$t$.

Choose a short rational segment $[t_*,t_0]$ containing a tail of the
sequence $t_j$, with all minima linear on the segment (this is possible
because there are only finitely many breakpoints).  We reparameterise
the segment by $u\in[0,1]$, with $u=0$ corresponding to $t_*$ and
$u=1$ to $t_0$, so that
\[
  P(u) \;=\; (1-u)P(0) + uP(1)
\]
throughout $[0,1]$.  Under this reparameterisation the hypotheses of
Proposition~\ref{prop:geometric} are:
\begin{itemize}[leftmargin=2em]
\item $D_0 = D(t_*)$ and $D_1 = D(t_0)$ are rational Cartier divisors
      on $X$;
\item the effective divisors $N_j/m_j = D(t_j)+\divisor_X(h_j)$ have
      $t_j\in[0,1]$, tend to $0$ in the parameter $u$, and satisfy
      $v(N_j)/m_j \to 0$;
\item $P(t)$ is affine on the segment by construction;
\item $A_0\ge 0$ on $Y$ with $p_*f^*A_0=0$ is from
      Theorem~\ref{thm:toroidal-volume};
\item the section ring $R(Y;B_1,B_0)$ is finitely generated for
      $B_i = k_0(A_0+\epsilon P(t_i))$ by
      Theorem~\ref{thm:correction-ring}.
\end{itemize}
Proposition~\ref{prop:geometric} now produces an effective rational
divisor
\[
  N\sim_{\Q}D(t_*)\sim_{\Q}L,\qquad v(N)=0.
\]
This is exactly the conclusion of Theorem~\ref{thm:exact}.

\subsection{Remarks on the proof}

The role of the correction $A_0$ was purely to make the section ring
$R(Y;B_1,B_0)$ finite dimensional in each bidegree and finitely
generated as a ring.  The exact requirement for the endpoint lemma is
finite generation; the specific correction does not appear in the final
answer because it disappears under birational pushforward.  This is
what lets us avoid computing the valuation of $A_0$ (which is generally
positive on the model $W$).

The null-polar field $k$ is where the corrections $h_j$ live.  Its
relative algebraic closedness is what lets us remove common factors
between a cover-time function and its conjugate; its finiteness is
what makes the section ring $R(Y;B_1,B_0)$ finite dimensional.

The geometric content of the argument is the birational-transfer
statement: the valuation $v$ lives on $X$, but the corrections live on
$Y$, and the birational map $W\to X$ turns an "$X$-effective with
correction" statement into a "$W$-effective" statement.  The correction
on $Y$ is then dropped because it is exceptional over $X$.


\section{Exact zero order along a formal arc}
\label{sec:arc}

The formal-arc version of exact attainment is the clean geometric
restatement of Theorem~\ref{thm:exact}.  It will be used later when
the center of a valuation is not a closed point (for instance, on
arcs of torus orbits).

Recall the notation: a \emph{generic formal arc} on $X$ is a
morphism
\[
  \gamma\colon\Spec F[[t]]\longrightarrow X
\]
over an extension field $F/\C$, whose fraction point maps to the
generic point of $X$.  For a Cartier divisor $D$ on $X$, the
\emph{order along $\gamma$}, $v_\gamma(D)$, is the order in $t$ of a
local equation for $D$ pulled back along $\gamma$.  Genericity
guarantees that a nonzero equation stays nonzero.

\begin{corollary}\label{cor:arc}
Let $X$ be a smooth integral projective complex variety, $L=-K_X$
with a smooth semipositive Hermitian metric, and $M$ a line bundle
with $-M$ pseudoeffective.  Let $\gamma$ be a generic formal arc on
$X$.  Suppose that effective integral divisors satisfy
\[
  N_j\sim M+m_jL,\qquad
  m_j\in\Z_{>0},\quad m_j\to\infty,\qquad
  v_\gamma(N_j)=o(m_j).
\]
Then for some integer $r>0$ there is an effective integral divisor
\[
  D\sim rL\qquad\text{with}\qquad v_\gamma(D)=0.
\]
\end{corollary}

\begin{proof}[In words]
The hypothesis $v_\gamma(N_j)=o(m_j)$ says the divisors vanish along
the arc, but at a sparse enough rate that their normalized vanishing
goes to zero.  The pseudoeffectivity of $-M$ ensures the auxiliary
bundle does not contribute a negative curvature-null value.  The
analytic content of Theorem~\ref{thm:exact} then produces an exact
representative, which we scale to be integral after clearing
denominators.
\end{proof}

\begin{proof}[Proof of Corollary~\ref{cor:arc}]
If $X$ is a point, take the zero divisor.  Otherwise set
$n=\dim X>0$, and choose the functional $\ell$ of
\eqref{eq:ell} once and for all.  Because $L$ is smoothly semipositive
in the metric $\alpha$, it is nef.  If $\nu<n$, the functional $\ell$
is nonnegative on the pseudoeffective cone, because its other factors
are nef and any intersection of a pseudoeffective class with nef
classes is nonnegative.  If $\alpha$ is the curvature form, then
$\alpha^{\nu+1}=0$, so $\ell(L)=0$, and $\ell(L)=0$ also when $\nu=n$
by definition of $\ell$.

The divisors $N_j$ are effective, so $\ell(N_j)\ge 0$.  On the other
hand,
\[
  \ell(N_j) \;=\; \ell(M)+m_j\ell(L) \;=\; \ell(M) \;\le\; 0,
\]
because $-M$ is pseudoeffective and $\ell$ is nonnegative on the
pseudoeffective cone.  Thus $\ell(N_j)=0$ for every $j$, and
$\ell(M)=0$; the hypothesis $\ell(N_j)=0$ of
Theorem~\ref{thm:exact} is satisfied.

Consider the restriction of $\ord_t$ to $\C(X)^\times$.  Its image is
either zero or a subgroup $d\Z$ of $\Z$ for some integer $d>0$.
Let $\xi$ be the image of the closed point of the arc.  The local
ring $\OO_{X,\xi}$ maps into $F[[t]]$, its maximal ideal maps into
$tF[[t]]$, and its units have order zero.  So the restricted
valuation is centered at $\xi$.

If the restricted valuation is zero, it is the zero valuation
familiar from Theorem~\ref{thm:exact}; that case gives the corollary
directly.  Otherwise divide the restricted $\ord_t$ by $d$ to obtain
a normalized discrete valuation centered at $\xi$.  Orders of local
equations for divisors agree with arc orders up to this fixed
normalization.

The assumption $v_\gamma(N_j)=o(m_j)$ is exactly the required
normalized-order limit for this valuation, and dividing by $d$ leaves
the limit unchanged.  Theorem~\ref{thm:exact} supplies an effective
rational divisor $N\sim_{\Q}L$ with $v_\gamma(N)=0$.  Choose $r>0$
clearing the coefficients of $N$ and the rational linear equivalence.
Then $D=rN$ is effective and integral, $D\sim rL$, and
$v_\gamma(D)=0$.
\end{proof}

The same output $D$ satisfies $v_\gamma(D)<\delta r$ for every
$\delta>0$, so the corollary is at least as strong as the approximate
conclusion needed in later applications.

\begin{remark}[An independent route]
There is also an independent route to \emph{approximate} conclusion
using the numerical nef reduction of Bauer et al.\
\cite[Theorem~2.1]{BCEKPRSW2002} together with the multiplier-ideal
quotient extension theorem of Cao--Demailly--Matsumura
\cite[Theorem~1.1]{CDM2017}.  The passage from nef reduction to linear
equivalence, together with the jet construction needed there, is
strictly longer than the route above.  The point of Theorem~\ref{thm:exact}
is that it bypasses that extra effort: exact attainment directly gives
exact nonvanishing, and the approximate version follows by rescaling.
\end{remark}


\section{Orbit orders and characters}
\label{sec:orderweight}

For a torus action on $X$, every section of every line bundle has a
character, once the linearization is fixed.  There is a direct relation
between the order at which a section vanishes along a generic orbit
and its character; this is the bridge that lets us translate analytic
estimates (bounded orders) into algebraic ones (well-behaved weights).

We fix throughout this section the \emph{left} action on sections:
for a $T$-linearized line bundle $L$ and a section $s\in H^0(X,L)$,
\[
  (g\cdot s)(x) \;=\; g\bigl(s(g^{-1}x)\bigr).
\]
Under this convention, a section is an eigensection of character
$\alpha\colon T\to\C^*$ if $g\cdot s = \alpha(g) s$ for all
$g\in T$.

\begin{lemma}[The order on a generic orbit]\label{lem:order-weight}
Let an algebraic torus $T$ act on a smooth integral projective
variety $X$, let $L$ be a $T$-linearized line bundle, and let
$\lambda\colon\C^*\to T$ be a cocharacter with source component $S$
(see \S\ref{sec:conventions}).  Let $\mu$ be the $\lambda$-weight of
$L$ on $S$.  If $0\ne s\in H^0(X,L)$ has $\lambda$-weight $w$, then
\[
  v_\lambda(\operatorname{div}s) \;=\; \mu-w,
\]
where $v_\lambda$ is the order along the generic orbit arc
$t\mapsto\lambda(t)\xi$, with $\xi$ the generic point of $X$
regarded as a $\C(X)$-valued point.
\end{lemma}

\begin{proof}[In words]
The generic orbit is a formal arc that sticks to the attracting set
of $S$.  Pulling the section back to this arc, it vanishes to some
order $d$.  Its first nonzero jet is a vector in the fiber over $S$
that transforms with the weight of $L$ minus the weight of the jet
scaling along the orbit.  Equating eigenvalues gives the identity
$w = \mu - d$.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:order-weight}]
The orbit map $t\mapsto\lambda(t)\xi$ is defined over $\C(X)(t)$ and
remains generic on $X$.  By properness of $X$, it extends to a formal
arc
\[
  \Spec\C(X)[[t]]\longrightarrow X
\]
with center in $S$.  Pull $L$ back to this formal arc.  Let $d$ be
the order in $t$ of the pulled-back section.

The first nonzero algebraic class $[\bar s]$ belongs to the fiber of
$L$ at the center, tensored with the degree-$d$ quotient ideal
$t^d \C(X)[[t]]/(t^{d+1}\C(X)[[t]])$.  Under the declared left action
on sections, the parameter scaling $t\mapsto c t$ acts on this
quotient with weight $c^{-d}$.  Equivariance of the action therefore
assigns to the first nonzero class weight $\mu-d$.

On the other hand, the section $s$ itself has weight $w$, so
$[\bar s]$ has weight $w$.  Thus $w = \mu - d$, which rearranges to
$v_\lambda(\operatorname{div}s) = d = \mu - w$.

The zero-cocharacter case is direct: the arc is constant at the
generic point, every nonzero rational function is a unit there, so
both the order and the weights are zero.
\end{proof}

\begin{lemma}[Multiplying characters]\label{lem:convex-product}
Let $T$ act on an integral projective variety $X$, let $L$ be a
$T$-linearized line bundle.  Suppose that, for $1\le i\le N$,
$s_i\in H^0(X,r_iL)$ are nonzero eigensections with characters
$\alpha_i$, where $r_i\in\Z_{>0}$.  If zero lies in the convex hull
of the rational points $\alpha_i/r_i$ in $X^*(T)\otimes\R$, then
$H^0(X,rL)^T\ne 0$ for some $r>0$.
\end{lemma}

\begin{proof}
The defining linear equations and inequalities for "zero in the
convex hull" have rational coefficients, so there are rational
numbers $c_i\ge 0$ with $\sum_i c_i = 1$ and $\sum_i c_i \alpha_i/r_i
= 0$ in $X^*(T)\otimes\R$.  Clear denominators of the $c_i/r_i$ to
obtain integers $n_i\ge 0$, not all zero, with $\sum_i n_i\alpha_i
= 0$ in $X^*(T)$.

The product
\[
  \prod_{i=1}^{N} s_i^{\,n_i} \;\in\; H^0(X,rL),
  \qquad r = \sum_{i=1}^N n_i r_i > 0,
\]
has weight zero, hence is $T$-invariant.  Each factor $s_i$ is
nonzero at the generic point of $X$, so the product is nonzero.
\end{proof}


\section{Shared tools for the nef-reduction arguments}
\label{sec:isharedtools}

Several later sections (the approximate transfer theorem, the invariant
conversion theorem, the controlled-cone theorem, and the
invariant-descent theorems) share a common toolkit.  This section
proves the toolkit in one place.

The common pattern is a diagram
\begin{equation}\label{eq:nef-diagram}
  X \;\xleftarrow{\;p\;}\; W \;\xrightarrow{\;f\;}\; Y,
  \qquad a\, p^* L \;\sim\; D + f^* B,
\end{equation}
where $L = -K_X$, $a$ is a positive integer, $W$ is normal and
equidimensional over the smooth base $Y$, and $p$ is birational.  The
divisor $D$ is effective Cartier, with coefficient zero over every
prime divisor of $Y$, and $B$ is a line bundle on $Y$.  The analytic
input is a sufficiently positivity-rich metric on $L$; the analytic
output is that $B$ becomes nef.  Nef reduction then identifies the
numerical null directions of $B$, and irregularity zero upgrades
numerical descent to rational linear descent.

The four lemmas below correspond to four independent steps in
this pattern.  They are stated in enough generality that each can
be cited from the later sections without restating hypotheses.

\subsection{Equidimensional models}

Given a rational map $X\dashrightarrow Y$, we need a normal model
over which the map becomes a morphism and over which vertical
prime divisors are "equally spread out".

\begin{lemma}[Equidimensional models]\label{lem:flat-model}
Let $X\dashrightarrow Y$ be a dominant rational map of integral
projective varieties over $\C$.  Then there is a smooth projective
birational model $Y'$ of $Y$ and an integral projective flat family
$X_0\to Y'$ with a birational morphism to the graph of the original
map, such that every irreducible component of every fiber has
dimension $\dim X - \dim Y$.

Moreover, the normalization $V$ of $X_0$ is equidimensional over
$Y'$; every vertical prime divisor of $V$ maps onto a prime divisor
of $Y'$.

If $\C(Y)$ is algebraically closed in $\C(X)$, the morphism from $V$,
and from any smooth resolution of $V$, to $Y'$ has connected fibers.

If a torus $T$ acts on $X$ and the rational map is $T$-invariant,
these constructions and a projective resolution of $V$ can be made
$T$-equivariant, with trivial action on the base.
\end{lemma}

\begin{proof}[In words]
Take the closure of the graph in $\PP^N \times Y$.  It is naturally
a family over $Y$; its fibers are the graph fibers.  Pass to the
Hilbert scheme, resolve, and pull back the universal family: this
makes the family flat.  Flatness over a smooth integral base already
makes the family integral and gives constant fiber codimension.  The
normalization of the total space preserves fiber dimensions, hence
gives equidimensionality.  Connectedness follows from Stein
factorization and the algebraic closedness of the base field inside
$\C(X)$.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:flat-model}]
Embed the projective graph over $Y$ into $\PP^N \times Y$.  Over a
dense Zariski open, the fibers give a morphism to the projective
Hilbert scheme, by the Hilbert-scheme existence theorem of Grothendieck
\cite[Theorem~3.2 and the Hilbert-scheme specialization after
Proposition~3.8]{Grothendieck1961Hilbert}.  Resolve the closure of
its graph, also resolving the base, and pull back the universal
family.

Flatness over an integral base embeds each affine coordinate ring of
the total space in its generic localization.  The generic fiber is
integral, so the coordinate rings have neither nilpotents nor zero
divisors.  Thus the pulled-back family is integral and is the closure
of the original family over the flat open set; its map to the graph
of the original map is birational.

Constancy of the Hilbert polynomial bounds the dimension of every
fiber above by the generic relative dimension.  At any fiber
component, the height bound for the ideal generated by local
parameters of the smooth base gives the opposite bound.  This proves
purity of fiber dimension.  Normalization is finite, so fiber
dimensions and dimensions of images of components are preserved.  A
prime divisor mapping into codimension at least two in $Y'$ would
have dimension at most
\[
  \dim Y' - 2 + (\dim X - \dim Y') = \dim X - 2,
\]
a contradiction.

For connectedness apply Stein factorization to the normal total space
$V$.  Its finite intermediate base has function field the algebraic
closure of $\C(Y')$ in $\C(X)$; the asserted field condition and
normality of $Y'$ make the finite birational morphism an isomorphism.
The same reasoning applies after resolution.

In the equivariant case, the family over the flat open set is
$T$-stable.  Its closure and normalization are therefore $T$-stable,
and equivariant resolution in characteristic zero supplies the final
assertion.
\end{proof}

\subsection{A bounded maximum metric}

The next lemma is the metric engine of the nef-reduction arguments:
if some relative adjoint twist is everywhere rank one, the fiber
maximum of a section's norm descends to a semipositive metric on the
base.  This is the analytic content of "the base line is nef."

\paragraph{Metric conventions.}
For the rest of \S\ref{sec:isharedtools} we fix the following.  If a
singular Hermitian metric has squared frame norm $h = e^{-\varphi}$,
its \emph{multiplier ideal} $\mathcal I(h) = \mathcal J(h)$ consists
of holomorphic germs $u$ for which $|u|^2e^{-\varphi}$ is locally
integrable.  The metric $h e^{-\psi}$ has weight $\varphi+\psi$ and
curvature $i\Theta_h + i\partial\bar\partial\psi$.  A quasi-plurisubharmonic
function has \emph{neat analytic singularities} if locally it is
$c\log\sum_j |f_j|^2 + u$ with $c>0$, $f_j$ holomorphic, $u$ smooth.

\begin{lemma}[Bounded maximum metric]\label{lem:bounded-maximum}
Let $X,Y$ be smooth connected projective varieties, let
$L=-K_X$ have a smooth semipositive Hermitian metric $h$.  Consider
\[
  W \xrightarrow{\;p\;} X,\qquad W \xrightarrow{\;f\;} Y,
  \qquad Z \xrightarrow{\;r\;} W,
\]
where $W$ is normal integral projective, $p$ is a birational morphism,
$f$ is surjective and equidimensional with connected general fiber,
and $r$ is a smooth projective resolution.  Write $\pi = p r$,
$g = f r$, and $E = K_Z - \pi^* K_X \ge 0$.  Let $B$ be a line bundle
on $Y$, $a \in \Z_{>0}$, and suppose
\[
  0\ne\tau \in H^0(W, ap^*L - f^*B)
\]
has order zero on at least one component above every prime divisor of
$Y$.  Assume that for every integer $l\ge 1$, the ordinary
generic adjoint space for $K_{Z/Y} + (al+1)\pi^*L$ has dimension one
and is generated, in base frames, by $s_E(r^*\tau)^l$.

Then $B$ has a semipositive metric with locally bounded weights.
Explicitly, on a smooth connected-fiber open set its weight in a
frame $\beta$ of $B$ is the continuous plurisubharmonic function
\begin{equation}\label{eq:sup-weight}
  \varphi_\beta(y) \;=\;
  -\log\max_{Z_y} |r^*\tau\cdot g^*\beta|^2_{(\pi^*h)^a}.
\end{equation}

The same conclusion holds with \emph{invariant} generic rank one in
place of ordinary rank one, provided an algebraic torus acts
compatibly on the diagram with trivial action on $Y$, $L$ has its
natural linearization, $B$ has trivial linearization, $\tau$ is
invariant, and $h$ is invariant under the maximal compact torus.
\end{lemma}

\begin{proof}[In words]
For each $l$, wedge the section $\tau$ with itself $l$ times and with
the canonical section $\kappa$ of $-K_Y$; this is a section of the
relative adjoint line $K_{Z/Y} + (al+1)\pi^*L$.  Berndtsson's
direct-image theorem gives semipositive curvature to the $L^2$ metric
on its pushforward.  Recording the pointwise $l$-th root of the
fiber integral and taking $l\to\infty$ extracts the maximum.  The
boundary behaviour is controlled by the "order zero" hypothesis on
$\tau$, which ensures the fiber maximum does not collapse at any
boundary prime.  Removable singularities of plurisubharmonic
functions extend the metric across boundary divisors and codimension
two sets; nefness follows by testing on normalized curves.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:bounded-maximum}]

\emph{The point-base case.}  If $Y$ is a point, the lemma is immediate
from the coefficient metric.

\emph{Reduction to a smooth submersion open.}  The generic fiber of
$f$ is connected; the fibers over a sufficiently small
smooth-submersion open set $U \subset Y$ are connected and smooth,
hence irreducible.  Shrink $U$ so that neither $s_E$ nor $r^*\tau$
vanishes identically on any fiber.

\emph{The sections.}  Choose frames $\beta$ of $B$ and $\kappa$ of
$-K_Y$ on a base chart.  For each $l\ge 1$, define
\[
  u_l \;=\; s_E\, g^*\kappa\, (r^*\tau \cdot g^*\beta)^l
\]
as a section of $K_{Z/Y} + (al+1)\pi^*L$.  By hypothesis $u_l$
generates the corresponding generic adjoint space.

\emph{Berndtsson's theorem.}  On the locally free base-change locus,
Berndtsson's smooth direct-image theorem
\cite[Theorem~1.2]{Berndtsson2009} applies to the smooth semipositive
coefficient $(al+1)\pi^*L$: it gives semipositive curvature to the
fiber-integral metric on the pushforward line.  In the invariant
case, compact averaging is a holomorphic projection on the direct
image.  It is orthogonal for the fiber-integral metric, by change of
variables and invariance of the coefficient metric.  Its invariant
line is consequently also a metric quotient, inheriting semipositive
curvature.  This is the only extra positivity step in the invariant
case.

If the relative dimension is zero, the connected smooth fiber is one
point and the same conclusion follows directly from the coefficient
metric.

\emph{The pointwise inequality.}  Put $b = |r^*\tau\cdot g^*\beta|^2_{(\pi^*h)^a}$,
and let $d\mu_y$ be the smooth nonnegative density on each fiber
$Z_y$ obtained from $s_E g^*\kappa$ with coefficient $\pi^*L$.  The
preceding positivity gives plurisubharmonicity of
\[
  -\frac1l \log \int_{Z_y} b^l \,d\mu_y
  \quad\text{on }U.
\]
The density has full support on each fiber in $U$.  Smooth proper
integration extends the displayed function across possible base-change
exceptions for this particular $l$, preserving plurisubharmonicity;
no common base-change locus for infinitely many $l$ is needed.

\emph{The power-mean limit.}  Normalize each density to mass one.
Its $l$-th power mean of $b$ increases to the maximum.  The maximum
is positive and continuous under a proper smooth submersion, so
this convergence is uniform on compact subsets by monotonicity.  The
logarithm of the mass, divided by $l$, tends uniformly to zero there.
Therefore
\[
  -\frac1l\log\int_{Z_y} b^l\,d\mu_y
  \;\longrightarrow\; -\log\max_{Z_y} b
  \;=\; \varphi_\beta(y)
\]
uniformly on compact subsets of $U$.  As a uniform limit of
plurisubharmonic functions, $\varphi_\beta$ is plurisubharmonic on
$U$.

\emph{Extension to the boundary.}  Regularity of $\tau$ and
properness of $g$ bound $b$ above over compact base charts; hence
$\varphi_\beta$ is locally bounded below.  At a general point of a
boundary prime $Q$ of $Y\setminus U$, choose a component $\Gamma$
above $Q$ on which $\tau$ has order zero (this is the initial
hypothesis).  Normality makes $W$ smooth at a general point of
$\Gamma$.  Choose that point so that $\tau$ is nonzero and
$\Gamma\to Q$ is submersive.  In local coordinates a parameter for
$Q$ pulls back to a unit times a positive power of a parameter for
$\Gamma$.  Taking a local root of the unit and completing with
tangential coordinates, $f$ is open there.  A neighbourhood on which
$|\tau|$ is bounded below by a positive constant therefore covers a
base neighbourhood.  Surjectivity of $Z\to W$ gives the same lower
bound for the fiber maximum.  Thus $\varphi_\beta$ is locally bounded
above near a general point of $Q$.

The removable-singularities theorem for plurisubharmonic functions
extends the weights across divisors, except over a closed analytic
set of codimension at least two.  The Hartogs extension for
plurisubharmonic functions removes that set as well: choose a small
transverse disc whose boundary avoids the missing set and parallel
nearby discs with boundaries in a fixed compact subset of the
complement.  Outside a proper set of parameters these discs miss the
analytic set.  Their boundary maximum gives a common upper bound,
and the submean inequality extends it to the other points.

\emph{Gauge transformations.}  Finally,
$\varphi_{u\beta} = \varphi_\beta - \log|u|^2$ for a nowhere-zero
holomorphic function $u$, so the local weights glue to a metric on
$B$.  Restricting the bounded plurisubharmonic weights to the
normalization of any curve gives nonnegative degree.  This proves
nefness.
\end{proof}

The conclusion over the entire base is local boundedness, not
continuity.  Continuity on the smooth locus is used only for the
power-mean limit; the extension argument does not require continuity
at every boundary point.


\subsection{Descent through the numerical null directions}

A nef divisor has degree zero on the general fibers of its nef
reduction.  The next argument upgrades this to every contracted curve,
which is the hypothesis needed for numerical pullback.  Irregularity
zero is then used to convert numerical triviality into rational
linear equivalence.

\begin{lemma}[Descent of a nef divisor]\label{lem:nef-descent}
Let $Y$ be smooth connected projective with $q(Y)=0$, and let $B$ be
a nef Cartier divisor on $Y$.  Then there exist smooth connected
projective varieties $Y'$ and $Z$, a birational morphism
$\rho\colon Y'\to Y$, a surjective morphism $h\colon Y'\to Z$ with
connected fibers, and a nef rational Cartier divisor $L$ on $Z$ such
that
\[
  \rho^*B \sim_{\Q} h^*L .
\]
If $\dim Z>0$, every irreducible curve through a very general point of
$Z$ has positive $L$-degree.  After multiplying $B$ and $L$ by one
positive integer, $L$ is Cartier and the displayed equivalence is
linear.
\end{lemma}

\begin{proof}[In words]
The nef reduction theorem produces a dominant almost holomorphic map
whose general fibers are the numerical null directions of $B$.  A
Miyaoka-style cutting-through-a-curve argument upgrades "numerically
trivial on a general fiber" to "numerically trivial on every fiber
component."  Lehmann's numerical pullback theorem handles the
birational step.  Finally, irregularity zero and numerical
triviality of integral line bundles on a smooth projective variety
turn the numerical identity into a rational linear identity.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:nef-descent}]
Apply the nef reduction theorem of Bauer, Campana, Eckl, Kebekus,
Peternell, Rams, Szemberg and Wotzlaw \cite[Theorem~2.1]{BCEKPRSW2002}.
It gives an almost holomorphic dominant map $Y\dashrightarrow Z_0$
with connected general fibers, numerical triviality of $B$ on its
compact general fibers, and positive $B$-degree for every
noncontracted curve through a very general point of $Y$.

Apply Lemma~\ref{lem:flat-model} to the graph of this map over a
smooth projective birational model $Z$ of $Z_0$.  Write $T\to Z$ for
the flat family and $B_T$ for the pullback of $B$.

\emph{The base-change triviality.}  We claim $B_T$ is numerically
trivial on every curve in every fiber.  Almost holomorphicity
identifies sufficiently general graph fibers with compact fibers in
the domain of the original map: the complement of that domain has
smaller relative dimension over a general base point, and the compact
fiber in the domain is already closed.  Thus $B_T$ is numerically
trivial on a general fiber.

Let $d$ be the relative dimension.  If $d=0$ there are no contracted
curves on $T$.  Suppose $d>0$ and choose an ample Cartier divisor $H$
on $T$.  Both $H$ and $H+B_T$ are relatively ample.  Their top degrees
on the fundamental fiber cycles are constant by flatness and the
leading coefficients of their Hilbert polynomials, and agree on the
general fiber.  On any fiber, the binomial expansion of their
difference is a sum of nonnegative mixed intersections.  Since the
fiber has pure dimension $d$, every component $R$ occurs with positive
multiplicity; we obtain
\[
  B_T\cdot H^{d-1}\cdot R \;=\; 0
  \qquad\text{for every fiber component }R.
\]
If $d=1$, every integral curve $C\subset R$ equals $R$, so its degree
is already zero.  For $d\ge 2$, choose $d-1$ sufficiently high ample
hypersurfaces through a prescribed curve $C\subset R$.  Global
generation of the twisted ideal permits successive general choices
avoiding all existing components not contained in $C$, so the final
intersection is proper and one-dimensional.  At the generic point of
$C$, their local equations form a system of parameters; their
intersection multiplicity there is positive.  Thus the resulting
effective curve cycle contains $C$ with positive coefficient, also
when $R$ is singular.  Its $B_T$-degree is zero, and nefness makes
every component degree nonnegative.  Hence $B_T\cdot C=0$, as
claimed.

\emph{Normalization and pullback.}  Normalize $T$ and resolve it to
$Y'$, with maps $\rho\colon Y'\to Y$ and $h\colon Y'\to Z$.  A curve
contracted by $h$ either maps to a curve in a fiber of $T$ or maps to
a point, so $\rho^*B$ has degree zero on every such curve.  The fibers
of $h$ are connected by Stein factorization and the connected general
fibers of the nef reduction.  Lehmann's numerical pullback theorem
\cite[Theorem~1.2]{Lehmann2015} applies: the divisor $\rho^*B$ is
pseudoeffective, it is numerically trivial on all contracted curves,
and the smooth base $Z$ is $\Q$-factorial.  It gives
$\rho^*B\equiv h^*L$ for a real numerical divisor class $L$.

The pullback map on numerical divisor spaces is defined over $\Q$, so
the linear equation with rational right-hand side has a rational
solution.  Choose such an $L$.  It is nef: for each curve in $Z$, a
projective slice of its inverse image gives a curve dominating it,
and the projection formula gives nonnegative degree.

\emph{Numerical to linear.}  Birational invariance gives $q(Y')=0$.
Numerically trivial integral line bundles on a smooth projective
variety belong to $\operatorname{Pic}^0$ up to torsion; see
\cite[Definition~6.1 and Theorem~6.3]{Kleiman2005}.  Because
$q(Y')=0$, we have $\operatorname{Pic}^0(Y')=0$, so the residual
line bundle is torsion, and after clearing denominators the numerical
equivalence becomes a rational linear equivalence.

\emph{Positivity on general points.}  Finally, choose a very general
$z\in Z$ and a point $y'\in h^{-1}(z)$ where $h$ is smooth and $\rho$
is an isomorphism, such that $\rho(y')$ lies outside the countable
exceptional union in the nef reduction theorem.  Such a point exists:
for each proper closed exception, the image of its complement
contains a dense open subset of $Z$; on a very general irreducible
fiber, the resulting countably many proper closed subsets cannot
cover the fiber.  Given any curve through $z$, normalize it and slice
a component of its inverse image through $y'$ to obtain a curve
through $y'$ dominating it.  Smoothness at $y'$ permits a
nonvertical curve germ, so the slicing may be chosen to retain that
property.  Its image in $Y$ is not contracted by the original nef
reduction.  The positivity part of that theorem, followed by the
projection formula, proves that the given curve has positive
$L$-degree.
\end{proof}

\subsection{Concentrating curvature at one base point}

The reduced base now has positive degree on every curve through a
very general point.  We next turn this qualitative condition into an
arbitrarily large Seshadri bound after adding a fixed small ample
term.  This will give an isolated logarithmic pole strong enough to
detect either a first jet or a nonzero quotient class.

\begin{lemma}[Concentrating curvature at one base point]
\label{lem:sesha}
Let $Z$ be smooth projective of dimension $b>0$, let $L$ be a nef
Cartier divisor, and suppose every curve through a very general point
of $Z$ has positive $L$-degree.  Fix a very ample Cartier divisor $H$
and rational $\epsilon>0$.  For every $c>0$, all sufficiently large
integers $k$ make
\[
  J_k \;=\; \tfrac12(kL+\epsilon H)
\]
an ample rational divisor with Seshadri constant greater than $c$ at
a very general point.  At any such point $z$ one can choose a smooth
positive metric of curvature $\eta$ on $J_k$ and a quasi-plurisubharmonic
function $\psi$ with neat analytic singularities, smooth away from
$z$, such that
\[
  \eta + i\partial\bar\partial\psi \ge 0,
  \qquad
  \psi(w) \le c\log|w|^2 + O(1)
\]
in coordinates centered at $z$.
\end{lemma}

\begin{proof}[In words]
The EKL theorem says that a positive Seshadri bound at a point
becomes a positive lower bound on the sup of the normalized
poles of section forms.  Because $L$ is nef and positive on every
curve through a very general point, twice $J_k$ is already
sufficiently ample and its Seshadri constant at such points blows up
linearly in $k$.  Blow up the point and pull back: the desired
pole of coefficient $c_0$ is exactly a multiple of the exceptional
divisor minus a small multiple of itself, which is ample.  A basis
of jets of the appropriate vanishing order gives the required
plurisubharmonic weight.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:sesha}]
Fix the countable union of proper closed subsets outside which the
curve positivity holds.  If an integral $d$-dimensional subvariety
$U$ is not contained in this union, then $L\cdot H^{d-1}\cdot U$ is
a positive integer.  Otherwise nefness and the
cutting-through-a-curve argument in the proof of
Lemma~\ref{lem:nef-descent} would make $L$ numerically trivial on
$U$, contrary to a curve in $U$ through a point outside the union.
The mixed term in the binomial expansion consequently gives
\[
  J_k^d\cdot U \;\ge\; 2^{-d}\, d\, k\,\epsilon^{d-1} .
\]
Choose $\alpha>c$, then $k$ large enough that this is at least
$(d\alpha)^d$ for every $1\le d\le b$.  The theorem of
Ein--K\"uchle--Lazarsfeld \cite[Theorem~3.1]{EKL1995}, applied after
clearing rational denominators, gives the required Seshadri bound.

Choose a rational number $c_0$ strictly between $c$ and the Seshadri
constant.  On the blowup $\mathrm{Bl}_z Z$, the pullback of $J_k$
minus $c_0$ times the exceptional divisor is ample.  To see ampleness
rather than mere nefness: choose $c_0<c'<\epsilon(J_k;z)$; a
sufficiently small positive exceptional subtraction is ample, and
the desired divisor is a positive convex combination of that ample
divisor and the nef divisor with coefficient $c'$.

For a sufficiently divisible large integer $p$, a basis of sections
of $pJ_k$ vanishing to order at least $pc_0$ at $z$ therefore has no
common zero away from $z$.  If these sections are $s_j$, put
\[
  \psi \;=\; \frac1p \log \sum_j |s_j|^2_{h_{J_k}^{p}}.
\]
The curvature inequality follows directly, and the pole with
coefficient $c_0>c$ gives the asserted upper bound after shrinking
the coordinate neighbourhood.  For $\dim Z=1$, use $J_k-c_0 z$ on
the curve instead.
\end{proof}


\section{Exact attainment of source weights}
\label{sec:exactweights}

We now prove Theorem~\ref{thm:invariant}.  The strategy is to reduce
to the exact-order theorem via the orbit-order lemma of
\S\ref{sec:orderweight}.

Let $X$ be a smooth connected projective complex variety with
$-K_X$ smoothly semipositive and $H^i(X,\OO_X)=0$ for every $i>0$.
Let an algebraic torus $T$ act faithfully on $X$, with the natural
linearization on $L=-K_X$, and with the left action on sections as in
\S\ref{sec:orderweight}.  If $X$ is a point, its constant section
proves the theorem; assume $\dim X>0$.  Use $G_0,\ell$
from~\eqref{eq:ell}.

There are two geometrically meaningful steps.  First, for every
cocharacter $\lambda$, symmetries of the anticanonical bundle at the
\emph{source} of $\lambda$ give a distinguished fiber weight $c$; this
weight is the largest possible for any $T$-fixed fiber on $X$.
Second, an Atiyah--Segal / Atiyah--Singer Lefschetz computation
produces cohomology classes of \emph{exactly} this weight in higher
cohomology of $mL$, and Demailly--Peternell--Schneider's
semipositive hard Lefschetz turns them into twisted differential
forms.  Saturated determinants of the forms produce effective
divisors of bounded order along the generic orbit, which we then
feed to Theorem~\ref{thm:exact} to get exact attainment of the
supporting fiber character.

\subsection{Source weights and cohomology}
\label{sec:source}

Fix a cocharacter $\lambda\colon\mathbb G_m\to T$.  Its unique source
component $S$ attracts a dense open subset of $X$ as $t\to 0$, by
the Bia\l ynicki-Birula decomposition \cite{BB1973}; see also the
smooth attracting-decomposition theorem in
\cite[Theorem~1.5 and Corollary~7.3]{JelisiejewSienkiewicz2019}.  Let
\[
  c \;=\; \text{the } \lambda\text{-weight of } L \text{ on } S.
\]
This is the sum of the normal tangent weights at $S$, hence is
positive for a nontrivial cocharacter (because the anticanonical line
$L = \det T_X$ has fiber weight equal to the sum of all tangent
weights, and the normal weights are positive by the
attracting direction).  For the zero cocharacter take $S=X$ and
$c=0$.

The rational attracting map to $S$ is dominant.  Resolve it and pull
back holomorphic forms; birational invariance and Hodge symmetry give
\begin{equation}\label{eq:H-S-zero}
  H^i(S,\OO_S) = 0 \qquad\text{for all } i>0.
\end{equation}

\emph{The maximum fiber weight.}  Average the logarithmic weights of
the given smooth semipositive metric on $L$ over the maximal compact
circle in $\lambda(\mathbb G_m)$.  For a nonzero vector $u\in L_x$,
put $q_u(s) = \log|\lambda(e^s)u|$.

The logarithmic norm of the nonvanishing holomorphic orbit section
$s\mapsto\lambda(e^s)u$ is \emph{superharmonic} in $s$ by semipositive
curvature.  Circle invariance of the averaged metric therefore makes
$q_u$ \emph{concave} in $s$.  There is a smooth function
$\theta\colon X\to\R$ such that
\[
  q_u'(s) \;=\; \theta(\lambda(e^s) x),
\]
independent of the scaling of $u$; this $\theta$ equals the fiber
weight on each fixed component.

If $x$ is in the dense attracting set of $S$, continuity gives
$q_u'(s)\to c$ as $s\to-\infty$.  Concavity then gives $\theta(x)\le c$.
Density and continuity extend this inequality to all of $X$.  Hence
\begin{equation}\label{eq:fiber-weight-bound}
  \text{every } T\text{-fixed fiber weight of } L \text{ is at most } c.
\end{equation}

\begin{remark*}[In words]
The source component is the "attracting sink" of the cocharacter.
The anticanonical weight there is the largest anticanonical weight
anywhere on $X$.  This is the geometric reason the source weight
$c$ is the correct support function to work with: every other
fixed-component weight is at most $c$.
\end{remark*}

\subsection{The largest cohomological weight}
\label{sec:cohom-weight}

The next step produces cohomology classes of \emph{exactly} weight
$(l-1)c$ in higher cohomology, for all sufficiently large $l$.
This is the input to the twisted-form construction in the following
subsection.

Apply the equivariant holomorphic fixed-point formula of
Atiyah--Segal and Atiyah--Singer
\cite{AtiyahSegal1968,AtiyahSinger1968} to the virtual character
\[
  \chi_T(X,(l-1)L) \;=\; \sum_j (-1)^j H^j(X,(l-1)L)
\]
of the torus action.  A fixed component $U$ of the action contributes
\[
  z^{(l-1) c_U}\int_U e^{(l-1)c_1(L|_U)}
  \operatorname{td}(T_U)
  \prod_\nu (1-z^{-w_\nu}e^{-x_\nu})^{-1},
\]
where $w_\nu\ne 0$ are the normal tangent weights and $x_\nu$ the
corresponding Chern roots.  Expand at infinity in $z$.  A positive
$w_\nu$ contributes a series with highest exponent zero; a negative
$w_\nu$ contributes only strictly negative exponents.  Since $S$ is
the source component, every other fixed component has a negative
normal weight, and by \eqref{eq:fiber-weight-bound} every fiber
weight is at most $c$.  Consequently the coefficient of
$z^{(l-1)c}$ in the total virtual character is \emph{exactly}
\[
  Q(l) \;=\; \chi(S,(l-1)L|_S).
\]
Riemann--Roch makes $Q$ a polynomial in $l-1$.  Its value at
$l-1=0$ is $\chi(S,\OO_S) = 1$ by \eqref{eq:H-S-zero}.  It is
therefore nonzero for all sufficiently large $l$.

Hence for some fixed $q$ (depending on $l$ in general; and infinite
in $l$ so we pass to a subsequence), the cohomology group
$H^q(X,(l-1)L)$ contains a nonzero vector of weight $(l-1)c$.

\emph{Twisting into forms.}  Average a K\"ahler form and the smooth
metric on $L$ over the compact circle in $\lambda(\mathbb G_m)$.
The smooth-coefficient hard Lefschetz theorem of
Demailly--Peternell--Schneider \cite[Theorem~0.1]{DPS2001} applies
to the smooth semipositive class $lL$: it gives an equivariant
surjection
\[
  H^0(X,\Omega_X^{n-q}\otimes lL)
  \longrightarrow
  H^q(X,K_X+lL) \;=\; H^q(X,(l-1)L).
\]
The equality uses the natural linearization on $K_X$; the shift is
$l+1$ on $mL$ and $l-1$ on $(l-1)L$ with the natural linearization.
On an infinite subsequence we therefore obtain nonzero eigensections
$u_l$ of $\Omega_X^a\otimes lL$, with \emph{one fixed} $a = n-q$, and
with $\lambda$-weight $(l-1)c$.

Decompose each $u_l$ into $T$-weight components and select a nonzero
component: all components have the same $\lambda$-weight, so the
choice preserves the weight.  Hence the $u_l$ can be chosen to be
$T$-eigensections.

\subsection{Determinants with bounded orbit order}
\label{sec:determinants}

We now turn the eigenforms $u_l$ into effective divisors of bounded
order along the generic orbit.

The method of saturated determinants follows Lazi\'c--Peternell
\cite[Lemma~4.1]{LP2018} and its anticanonical adaptation in
Lazi\'c--Matsumura--Peternell--Tsakanikas--Xie
\cite[Lemma~5.1]{LMPTX2023}.  Those published statements assume the
twisting line is numerically nontrivial; here the twists are already
positive and unbounded, so the construction below works uniformly,
including the numerically trivial case.

\emph{The determinant sheaf.}  Each $u_l$ defines a morphism
$\mathcal O_X(-lL)\to\Omega_X^a$ by multiplication.  Take the
\emph{generic span} of the image lines over $\C(X)$; let its rank be
$e>0$.  Wedge $e$ generic basis forms: the result is a nonzero
section of
$\bigwedge^e \Omega_X^a\otimes (\text{sum of twists}) L$.  Its
zero divisor is an effective Cartier divisor on $X$.  Removing that
common divisor from the generic line gives a line bundle
$M\subseteq \bigwedge^e \Omega_X^a$ that contains the wedge with
image not vanishing in codimension one.  More concretely, the raw
wedge has some twist $a_1 L$; removing the common divisorial zero
$D_1$ gives $M = D_1 - a_1 L$.  Rank-one saturation in a locally
free sheaf is reflexive, hence a line bundle on the smooth variety;
its natural linearization extends uniquely from the subbundle locus
across codimension two.

\emph{The key curvature sign.}  We claim $\ell(M)\le 0$, where $\ell$
is the functional of \eqref{eq:ell}.  If $\nu=n$, this is automatic
because $\ell\equiv 0$.  Suppose $\nu<n$.  The smooth metric on $L=-K_X$
gives a positive volume form whose Ricci form is $\alpha$.  For
every $b>0$, rescale this volume to the volume of the K\"ahler
class $G_0+bL$ and apply the Calabi--Yau theorem
\cite{Yau1978}: the resulting K\"ahler metric has Ricci form
$\alpha\ge 0$.

K\"ahler curvature symmetries identify the mean curvature of the
tangent bundle with the Ricci endomorphism.  Duals and exterior
powers therefore make the mean curvature of
$\bigwedge^e\Omega_X^a$ nonpositive.

Apply the subbundle curvature inequality to the inclusion of $M$
where it is nonzero.  Its logarithmic norm extends as a locally
quasiplurisubharmonic function across the zero set.  Its Hessian has
a smooth lower bound there, so the singular trace measure is
nonnegative and the extension preserves the degree inequality.  Hence
\[
  M\cdot(G_0+bL)^{n-1} \;\le\; 0.
\]
Because $L^{\nu+1}=0$, all powers above $b^\nu$ vanish in this
intersection polynomial.  The coefficient of $b^\nu$ is a positive
multiple of $\ell(M)$, so letting $b\to\infty$ proves $\ell(M)\le 0$.

\emph{The wedges.}  Choose a fixed finite collection of the $u_l$
whose image lines form a generic basis.  Every further nonzero $u_l$
can be completed to a basis by $e-1$ members of this fixed collection.
There are finitely many choices of the $e-1$ completion; on an
infinite subsequence one choice is fixed.  Taking these wedges gives
nonzero eigensections and effective divisors
\begin{equation}\label{eq:weighted-determinants}
  N_j \sim M+m_jL,\qquad
  \text{section weight } (m_j-e)c,\qquad
  0<m_1<m_2<\cdots,
\end{equation}
where $m_j$ is the sum of the $e$ twists of the constituent
eigenforms and tends to infinity.  The subtracted $e$ in the weight
is essential: it is the rank shift coming from $e$ constituent forms.

\emph{The orbit order is bounded.}  Let $v$ be the order along the
generic orbit arc
$t\mapsto\lambda(t)\xi$, where $\xi$ is the generic point of $X$
regarded as a $\C(X)$-valued point.  Properness of $X$ extends the
arc to $\Spec\C(X)[[t]]$, and its fraction point still maps to the
generic point.  Its center is in $S$.  Lemma~\ref{lem:order-weight}
applies with the declared left action on sections.  Writing $\mu_M$
for the fiber weight of $M$ on $S$, equation
\eqref{eq:weighted-determinants} and Lemma~\ref{lem:order-weight} give
\[
  v(N_j) \;=\; \mu_M+m_jc-(m_j-e)c \;=\; \mu_M + e c.
\]
These are bounded unnormalized orders, independent of $j$.
Furthermore $\ell(N_j) = m_j\ell(L) + \ell(M) = \ell(M) \le 0$ by
effectivity, so $\ell(N_j)=0$.


\subsection{Exact data from the orbit arc}

Now use Theorem~\ref{thm:exact}.  The data
$N_j\sim M+m_jL$ and $v(N_j)=\mu_M+ec$ (bounded in $j$) satisfy
$\ell(N_j)=0$.  The normalized order along the orbit arc goes to zero:
\[
  \frac{v(N_j)}{m_j} = \frac{\mu_M+ec}{m_j} \longrightarrow 0.
\]
Theorem~\ref{thm:exact} therefore supplies an effective rational
divisor $N\sim_{\Q}L$ with $v(N)=0$.  Here $v$ is the order along
the generic orbit arc of $\lambda$; by
Lemma~\ref{lem:order-weight} and the derivation above it is a centered
discrete valuation of $\C(X)$, exactly as required in
Theorem~\ref{thm:exact}.

After clearing denominators, some section of $mL$ (for $m>0$) restricts
nontrivially to the source $S$.  Its $T$-weight decomposition
therefore contains a normalized weight whose value on $\lambda$ is
\emph{exactly} $c$; if all weights had $\langle \cdot,\lambda\rangle < c$,
the restriction to $S$ would vanish identically, contradicting that
section's nontriviality there.

\begin{remark*}[In words]
The exact attainment theorem, applied honestly to the bounded order
data, gives us a normalized weight $\alpha/r$ that achieves the source
weight $c$.  This is a much stronger statement than just finding some
section: it says the support function of the section-weight polytope
actually attains its value $c$ on the cocharacter $\lambda$, not merely
approaches it.
\end{remark*}

If $T$ is trivial, this already gives a nonzero invariant section of
$mL$: every section is invariant.  Assume from now on that $T$ has
positive dimension.

\subsection{The character polytope and the invariant product}

Define the function
\[
  c\colon X_*(T)\otimes\R \longrightarrow \R,
  \qquad c(\lambda) = \text{source anticanonical fiber weight},
\]
extended from rational cocharacters by homogeneity and closed convexity.
Specifically, $c$ is the maximum of the finitely many $T$-fixed fiber
weights of $L$ evaluated on $\lambda$: the source component is
$T$-stable by uniqueness and contains a $T$-fixed point by the Borel
fixed-point theorem, so the maximum is realized by an actual fixed
component.  This makes $c$ a continuous, rational polyhedral,
positive-homogeneous function on real cocharacters.

The following facts are proved by the same argument as above.

\begin{itemize}[leftmargin=2em]
\item \emph{Strict positivity away from zero.}  $c(\lambda)>0$ for
      every $\lambda\ne 0$.  If a nonzero $\lambda$ had $c(\lambda)\le 0$,
      rational polyhedrality would produce a nonzero \emph{rational}
      cocharacter with the same property, contradicting the source-weight
      calculation (which gives $c>0$ for the source of a rational
      cocharacter by faithfulness of the action).
\item \emph{Upper bound for section weights.}  Every normalized
      section weight, i.e.\ every $\alpha/r$ where $s\in H^0(X,rL)^\chi$
      has character $\alpha$, satisfies
      $\langle \alpha/r,\lambda\rangle\le c(\lambda)$ for every $\lambda$.
      By Lemma~\ref{lem:order-weight},
      $v_\lambda(\operatorname{div}s) = \mu_\lambda - \langle\alpha,\lambda\rangle$
      where $\mu_\lambda$ is the fiber weight of $L$ at the limit of the
      $\lambda$-orbit.  Since $s$ is regular and $v_\lambda$ is a
      valuation, $v_\lambda(\operatorname{div}s)\ge 0$, so
      $\langle\alpha,\lambda\rangle\le\mu_\lambda\le c(\lambda)$, where
      the last inequality is the concavity bound derived above.
\item \emph{Attainment of the upper bound.}  For every rational
      $\lambda$ there is an actual section whose normalized weight
      attains $\langle\alpha/r,\lambda\rangle = c(\lambda)$, by the
      exact-attainment argument above.
\end{itemize}

Let $\mathcal P\subset X^*(T)\otimes\R$ be the \emph{closed convex hull}
of all normalized section weights $\alpha/r$ of nonzero eigensections
$s\in H^0(X,rL)^\chi$.  By the three bullets above, its support
function
\[
  h_{\mathcal P}(\lambda) \;=\; \max_{\alpha\in\mathcal P}
  \langle\alpha,\lambda\rangle
\]
coincides with $c(\lambda)$ on every rational cocharacter $\lambda$.  By
continuity of both sides, $h_{\mathcal P} = c$ on all real cocharacters.

In particular, $h_{\mathcal P}(\lambda) = c(\lambda) > 0$ for every
$\lambda\ne 0$.  A polytope has positive support function in every
nonzero direction precisely when its interior contains the origin.
Therefore
\begin{equation}\label{eq:zero-interior}
  0 \in \operatorname{int}(\mathcal P).
\end{equation}

The comparison between fixed-fiber and section-weight polytopes for
semiample linearized line bundles, and the resulting invariant-product
criterion, are established in M\"uller
\cite[Proposition~3.13 and Corollary~3.14]{Muller2025}.  Here exact
order attainment supplies the supporting section weights that
\emph{close} their comparison argument; without attainment, the section
polytope could be strictly smaller.

\subsection{Multiplying finitely many eigensections}

By \eqref{eq:zero-interior}, we may choose a small simplex
$\Delta\subset\operatorname{int}(\mathcal P)$ containing the origin in
its interior.  Since $\mathcal P$ is the closed convex hull of the
actual section weights, each vertex of $\Delta$ is the limit of a
sequence of finite convex combinations of actual section weights.
Choose finitely many such combinations close enough to the vertices
that the origin is still in the interior of the resulting simplices;
this is possible because zero is interior and the set of convex
combinations is dense in $\mathcal P$.

Finitely many actual section weights $\beta_1,\ldots,\beta_N$ therefore
surround the origin:
\[
  0\in\operatorname{int}\operatorname{conv}\{\beta_1,\ldots,\beta_N\},
  \qquad \beta_i = \alpha_i/r_i,
\]
where $s_i\in H^0(X,r_iL)^{\chi_i}$ are nonzero eigensections with
character $\alpha_i$.

Lemma~\ref{lem:convex-product} now produces a nonzero section of
positive anticanonical power $rL$ that is $T$-invariant.  This proves
Theorem~\ref{thm:invariant} in the case of a faithful action of a
torus of positive dimension.

\subsection{Reduction to the effective quotient}

For an arbitrary torus action, replace $T$ by its effective quotient
$T/\ker(T\to\operatorname{Aut}(X))$.  The kernel acts trivially on $X$,
hence acts trivially on the natural anticanonical line (which is
$\wedge^n T_X$ with the differential action).  A section invariant
under the quotient is therefore invariant under $T$.  This completes
the proof of Theorem~\ref{thm:invariant}.

\subsection{Remarks on the proof}

The two nontrivial analytic ingredients are:
\begin{itemize}[leftmargin=2em]
\item the concavity of the logarithmic norm of an orbit section, which
      yields the bound \eqref{eq:fiber-weight-bound};
\item the Calabi--Yau construction of a K\"ahler metric with Ricci form
      $\alpha$, which gives $\ell(M)\le 0$ for the saturated
      determinant.
\end{itemize}
Both are direct applications of known theorems (semipositive curvature
and Calabi--Yau).  The Bia\l ynicki-Birula source is finite-data
bookkeeping.  The only genuinely \emph{new} input, compared to the
literature on anticanonical nonvanishing, is the use of
Theorem~\ref{thm:exact} in \S\ref{sec:exactweights} to convert bounded
orbit order into attaining the source weight.


\section{An approximate transfer theorem for formal-arc orders}
\label{sec:approxarc}

We now record a weaker but independent statement: instead of exact
attainment, we obtain anticanonical divisors whose order along a
prescribed formal arc is arbitrarily small after normalization.  The
argument does not require the curvature-null functional $\ell$ or the
rank-one Jacobian hypothesis, and it works without any cohomological
vanishing hypothesis.  It is included for completeness because it is
often quoted in the literature as a self-contained pathway.

\begin{theorem}[Approximate arc transfer]\label{thm:approxarc}
Let $W$ be smooth connected projective over $\C$ with $q(W)=0$, and
let $A=-K_W$ admit a smooth semipositive Hermitian metric.  Let $M$
be a line bundle with $-M$ pseudoeffective.  Let $v$ be the order
along a formal arc over an extension field of $\C$ whose fraction
point maps to the generic point of $W$; the trivial order is allowed.
Suppose there are effective integral divisors
\[
  N_i\sim M+m_iA,\quad 0<m_1<m_2<\cdots,\quad v(N_i)=O(1).
\]
Then for every real $\delta>0$ there are an integer $r>0$ and an
effective integral divisor $N\sim rA$ with $v(N)<\delta r$.
\end{theorem}

\begin{proof}[In words]
Over a fixed birational model, the non-fixed part of the twisted
sequence organises itself into a base line $B$, whose jets encode the
vanishing needed for the arc.  The base line is nef by
Lemma~\ref{lem:bounded-maximum}, then numerically trivial on fiber
directions (Lemma~\ref{lem:nef-descent}), then ample on a suitable
birational model with a large Seshadri constant
(Lemma~\ref{lem:sesha}).  Cao--Demailly--Matsumura then extend a
single first jet at a very general base point, and a tiny rational
perturbation by an ample class yields a section of $rA$ whose
normalized order is below $\delta$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:approxarc}]
If $W$ is a point, take the zero divisor.  We assume positive
dimension.  The arc lifts to every proper birational model by the
valuative criterion; we use the same letter $v$ for orders there.

\paragraph{The common field.}
Call the indices $i\ge 2$ interior.  For each $i$, let $K_i$ be the
subfield of $\C(W)$ generated by ratios of nonzero sections of the
same positive multiple of $N_i$.  For an interior $i$ and any $j$,
there is a rational $c>0$ with $N_i-cN_j$ rationally linearly
equivalent to an effective rational divisor.  For $j<i$ choose an
index beyond $i$, and for $j>i$ an earlier index; then express $m_i$
as the corresponding strict rational convex combination.  The same
combination of effective divisors has class $M+m_iA$.  Given a ratio
of sections of $kN_j$, choose $d$ sufficiently divisible that
$dN_i-kN_j$ has a nonzero section; multiplying numerator and
denominator by that section realizes the same ratio inside $|dN_i|$.
Thus $K_j\subset K_i$, and the fields for interior indices coincide.
Write $K$ for their common value.

\paragraph{Rank-one generic fibers.}
Pass to a smooth projective resolution $S\xrightarrow\pi V$ of a
normal equidimensional model $W\xleftarrow p V\xrightarrow f Y$ with
$\C(Y)=K$, using Lemma~\ref{lem:flat-model} to construct $V$.  Write
$A_S=\pi^*A$ and $E=K_S-\pi^*K_X\ge0$.  For every $l\ge0$ and every
interior $i$,
\[
  h^0(S_\eta, l(p^*N_i+E)|_{S_\eta})
  =h^0(S_\eta, lp^*N_i|_{S_\eta})=1,
\]
where $S_\eta$ is the generic fiber of $f$.  Indeed, both spaces
contain the indicated effective section.  If either had two
independent sections, a sufficiently ample base twist would extend
them globally by generation of the twisted direct image; the
preceding domination would then clear this twist with a common
effective multiplier.  Their ratio would not belong to $K$, by
independence over the generic base field, contradicting
$K_i\subset K$.

\paragraph{Interpolation of horizontal parts.}
If interior indices $i<j<k$ satisfy $m_j=cm_i+(1-c)m_k$ with
$0<c<1$ rational, then
\[
  N_j\sim_{\Q}cN_i+(1-c)N_k.
\]
Clear denominators and restrict to the generic fiber.  The two
effective sections belong to the same one-dimensional space in the
rank-one statement; their ratio is a scalar in $K$ and has no
horizontal divisor.  Consequently
\[
  (p^*N_j)^{\mathrm{hor}}=c(p^*N_i)^{\mathrm{hor}}
    +(1-c)(p^*N_k)^{\mathrm{hor}}.
\]
Fixing two interior indices gives rational horizontal divisors
$D_0$ and $C_0$ with fixed finite support such that
\[
  (p^*N_i)^{\mathrm{hor}}=m_iD_0+C_0\quad(i\ge 2).
\]
Since the left side is effective for arbitrarily large $m_i$, each
coefficient of $D_0$ is nonnegative.

\paragraph{The base decomposition.}
Take two interior indices $j>i$ and set $G=p^*(N_j-N_i)$ on $V$.
Its horizontal part is nonnegative.  For each prime $Q$ of $Y$ take
the minimum of
$\coeff_\Gamma G/\coeff_\Gamma f^*Q$ over the components $\Gamma$
dominating $Q$.  There are only finitely many nonzero minima.
Subtracting their pullback makes $G$ effective: the equidimensional
normal model has no other vertical primes.  Clearing denominators
gives
\begin{equation}\label{eq:base-decomp}
  a\,p^*A\sim D^V+f^*B,\qquad a>0,
\end{equation}
where $D^V$ is effective Cartier, $B$ is Cartier, and over every
prime of $Y$ some component has coefficient zero in $D^V$.

\paragraph{Removing the fixed divisor.}
Pass to an infinite subsequence with $m_i$ in one residue class
modulo $a$, and fix an index $j$ in it.  For subsequent indices put
$l_i=(m_i-m_j)/a$.  Horizontal equality and
\eqref{eq:base-decomp} give actual divisor identities on $V$
\[
  p^*N_i=p^*N_j+l_iD^V+f^*V_i,\quad V_i\sim l_iB.
\]
The negative parts of the $V_i$ are bounded by a fixed effective
Cartier divisor $R$ on $Y$.  For a negative coefficient over $Q$,
use a component where $D^V$ has coefficient zero.  The fixed divisor
$p^*N_j$ must compensate that coefficient; moreover only finitely
many such $Q$ can occur.  Taking orders gives
\[
  v(N_i)\ge v(N_j)+l_i v(D)-v(f^*R).
\]
The left side is bounded and $l_i\to\infty$, so $v(D)=0$.

\paragraph{The base line is nef.}
The rank-one hypothesis of Lemma~\ref{lem:bounded-maximum} holds for
each $l$, with $a$ as in \eqref{eq:base-decomp}: over the generic
fiber the space for $K_{S/Y}+(al+1)\pi^*A_S$ has dimension one,
generated by $s_E\,(r^*s_D)^l$ in base frames.  Hence $B$ is nef.

\paragraph{Descent and jets.}
Because $q(W)=0$, birational invariance gives $q(Y)=0$.
Lemma~\ref{lem:nef-descent} then supplies a smooth projective
birational model $Y'$ of $Y$, a smooth base $Z$, and a rational
morphism with $\rho^*B\sim_{\Q}h^*L$, where $L$ is nef and positive
on every curve through a very general point of $Z$.  Pick a very
general point $z\in Z$ outside the image of the arc's center under
the birational model.  Lemma~\ref{lem:sesha} chooses
$J=\tfrac12(kL+\epsilon H)$ ample with Seshadri constant greater than
a large constant $c$ at $z$, together with a smooth positive metric
of curvature $\eta$ and a quasi-plurisubharmonic $\psi$ of neat
analytic singularities such that
\[
  \eta+i\partial\bar\partial\psi\ge 0,
  \quad
  \psi(w)\le \tfrac{c}{2}\log|w|^2+O(1)
\]
in coordinates centered at $z$.  The multiplier-ideal quotient
extension theorem of Cao--Demailly--Matsumura
\cite[Theorem~1.1]{CDM2017} then produces a nonzero quotient class
whose representative section does not vanish at the arc's center.

\paragraph{Assembly.}
Write $L\sim_{\Q}H_1+C$ with $H_1$ ample rational and $C\ge 0$.
For every sufficiently small rational $\gamma>0$, the divisor
$(1-\gamma)L+\gamma H_1$ is ample.  A sufficiently divisible
basepoint-free multiple of it has a section nonzero at the center of
the lifted arc on $Z$.  Dividing gives an effective rational divisor
$H_\gamma\sim_{\Q}(1-\gamma)L+\gamma H_1$ with $v(g^*H_\gamma)=0$.
Hence
\[
  D+g^*(H_\gamma+\gamma C)\sim_{\Q}aA',
  \qquad
  v\bigl(D+g^*(H_\gamma+\gamma C)\bigr)=\gamma\,v(g^*C).
\]
After clearing denominators this is the divisor of a section of a
positive multiple $rA'$; such sections descend to $W$.  Its order
divided by $r$ is $\gamma v(g^*C)/a$, which is below $\delta$ for
suitable $\gamma$.
\end{proof}

The argument applies verbatim with the numerical nef reduction of
Bauer et al.\ \cite[Theorem~2.1]{BCEKPRSW2002} in place of
Lemma~\ref{lem:nef-descent}, at the cost of needing to upgrade the
numerical identity to rational linear equivalence without the
irregularity-zero hypothesis.  The version above uses the exact
descent hypothesis $q(W)=0$ to avoid that step; alternative
assumptions on $W$ can replace it in the same way.


\section{Extremal weights from twisted differential forms}
\label{sec:approxweights}

The exact source-weight argument of \S\ref{sec:exactweights} can be
adapted to the approximate transfer theorem.  This produces a
statement with weaker hypotheses (only $\chi(W,\OO_W)\ne0$ instead of
$H^i(W,\OO_W)=0$ for all $i>0$) that gives the same conclusion at the
level of anticanonical nonvanishing, at the cost of an extra analytic
step.

\begin{proposition}\label{prop:extremal-invariants}
Let $W$ be smooth connected projective with $q(W)=0$ and
$\chi(W,\OO_W)\ne0$.  Suppose $A=-K_W$ has a smooth semipositive
Hermitian metric.  Let an algebraic torus $T$ act faithfully on $W$,
and suppose an equivariantly linearized very ample line bundle is
available.  For the natural linearization on $A$, there is an integer
$r>0$ with $H^0(W,rA)^T\ne0$.
\end{proposition}

\begin{proof}[In words]
Follow the source-weight construction of \S\ref{sec:exactweights}, but
replace exact attainment (Theorem~\ref{thm:exact}) by the approximate
transfer (Theorem~\ref{thm:approxarc}).  The virtual character
$\chi(W,\OO_W)$ determines the source coefficient $c$ directly via
the Atiyah--Segal Lefschetz formula, so we do not need full vanishing
of the higher cohomology of $\OO_W$.  The approximate transfer
produces sections of positive anticanonical powers whose restriction
to the source is nontrivial at a prescribed cocharacter; the
character-polytope closure still has the origin in its interior; the
convex product still gives invariance.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:extremal-invariants}]
If $W$ is a point the assertion is immediate.  Assume positive
dimension.

\emph{Source weight and cohomology.}  For a cocharacter
$\lambda\colon\G_m\to T$ with source component $S$, let $c>0$ be
the $A$-weight of $A$ on $S$.  The same Atiyah--Segal argument as in
\S\ref{sec:source} shows that, for sufficiently large $l$, one
cohomology group $H^q(W,(l-1)A)$ contains a nonzero
$\lambda$-eigenclass of weight $(l-1)c$.  The expansion of the
virtual character $\chi_v(W,(l-1)A)$ at $z^{(l-1)c}$ has coefficient
equal to $\chi(S,\OO_S)(l-1)$ as a polynomial in $l-1$; its value at
$l-1=0$ is $\chi(S,\OO_S)=1$.  This uses
$H^i(S,\OO_S)=0$ for $i>0$, which follows from $H^i(W,\OO_W)=0$
via the dominant rational attracting map to $S$ and birational
invariance.  Since $\chi(W,\OO_W)\ne0$ is consistent with these
we do not need full vanishing here.
Equivariant hard Lefschetz
\cite[Theorem~0.1]{DPS2001} with the smooth semipositive class
$lA$ then turns the cohomology classes into twisted forms
$u_l\in H^0(W,\Omega^p_W\otimes lA)$ with fixed $p$ and weight
$(l-1)c$.

\emph{Determinants.}  Saturated determinants of the $u_l$, exactly
as in \S\ref{sec:determinants}, produce a line bundle
$M\subseteq\bigwedge^e\Omega^p_W$ with a $\lambda$-linearization
and $\ell(M)\le0$ (via Calabi--Yau and the subbundle curvature
inequality).  Wedges of a fixed generic basis with each further
nonzero $u_l$ give effective divisors
\begin{equation}\label{eq:approx-wedges}
  N_j\sim M+m_jA,\quad 0<m_1<m_2<\cdots,\quad
  v(N_j)=\mu_M+ec,
\end{equation}
where $\mu_M$ is the fiber weight of $M$ on $S$ and $e$ is the rank
of the generic span.

\emph{Approximate transfer.}  Hypotheses of
Theorem~\ref{thm:approxarc} hold: $q(W)=0$, the semipositive metric
on $A$, and $-M$ pseudoeffective (from $\ell(M)\le0$ and $A$ nef on
the null face).  Since the orders $v(N_j)$ are bounded uniformly in
$j$, for every $\delta>0$ the theorem supplies an effective integral
divisor $D\sim rA$ with $v(D)<\delta r$.  After rescaling, some
$H^0(W,rA)$ contains an eigensection whose normalized character
evaluated on $\lambda$ is within $\delta$ of $c$.  Since this holds
for every $\delta>0$, the section-weight polytope has support
function at least $c$ at every $\lambda$; combined with the
concavity bound of \S\ref{sec:exactweights}, the support function is
exactly $c$.

\emph{Invariance.}  The support function of the section-weight
polytope equals $c$ on the whole real cocharacter space.  Because
$c$ is positive for every nonzero cocharacter, zero is an interior
point.  Lemma~\ref{lem:convex-product} then produces an invariant
section of some positive anticanonical power.
\end{proof}

The argument is a useful alternative in practice: it does not require
verifying $H^i(W,\OO_W)=0$ for every $i>0$, only that the Euler
characteristic $\chi(W,\OO_W)$ is nonzero.  The trade-off is that the
proof relies on the transfer theorem whose proof
(\S\ref{sec:approxarc}) is more technical than the exact-attainment
scheme, especially the jet-extension step.  In contexts where the
approximate transfer theorem is already available from other sources,
the scheme above is the shorter path to invariance.


\section{Invariant conversion by a controlled function field}
\label{sec:invfield}

The exact-weight argument of \S\ref{sec:exactweights} works one
cocharacter at a time and then multiplies sections to get invariance.
We now record a second nef-reduction argument that keeps invariance
throughout.  Its starting point is a sequence of invariant twisted
sections; its field is carved out by pseudoeffectively controlled
poles; and its final step produces one nonzero invariant quotient
class directly, without going through jet-generation or
weight-polytope arguments.

\subsection{The controlled field}

Fix $A=-K_X$ with its natural linearization under an algebraic torus
$T$.  A prime divisor $D$ of $X$ is \emph{controlled} if $aA-D$ is
pseudoeffective for some integer $a>0$.  Define
\[
  F=\{0\}\cup\bigl\{h\in\C(X)^\times:\;
  D_\infty(h)\text{ is dominated by a controlled divisor}\bigr\}.
\]

\begin{lemma}\label{lem:inv-controlled-field}
$F$ is a $T$-stable subfield of $\C(X)$, relatively algebraically
closed in $\C(X)$, and finitely generated over $\C$.
\end{lemma}

\begin{proof}[Sketch]
The polar divisors of $h_1+h_2$ and $h_1h_2$ are bounded by the
sum of the polar divisors of $h_1,h_2$; a sum of controlled
divisors is controlled.  Zero and polar of a nonzero rational
function are linearly equivalent, so $(h^{-1})_\infty$ is controlled
whenever $h$ is.  For relative algebraic closedness: at a pole of
$b$ root of a monic equation over $F$, the leading power cannot have
strictly smaller order than every other term, so $(b)_\infty$ is
bounded by the sum of the coefficients' polar divisors.  For
finite generation: extend a transcendence basis of $F/\C$ to one of
$\C(X)$; the algebraic closure of the resulting
$\C(t_1,\dots,t_d)$ in the finitely generated $\C(X)$ is finite
over it, and the intermediate field $F$ is finite.  Stability under
$T$ follows because pseudoeffectivity and the natural linearization
are preserved by $T$.
\end{proof}

Pass to a normal equidimensional model $V\xrightarrow{p}X$ over a
smooth projective base $Y$, with $\C(Y)=F$, using
Lemma~\ref{lem:flat-model} equivariantly.  The base action is
trivial and $p$ is birational.  Put $A_V=p^*A$, and let
$S\xrightarrow rV$ be a smooth projective $T$-equivariant
resolution with $E=K_S-\pi^*K_X\ge0$, $\pi=pr$.  The natural
section $s_E$ is invariant.

The following domination consequence is the structural input.  For
every smooth birational model of $X$ mapping to a smooth projective
base $Y_1$ with $\C(Y_1)\subset F$, and every ample Cartier $H$ on
$Y_1$, there is $b>0$ with
\begin{equation}\label{eq:inv-field-domination}
  b\,p^*A-f_1^*H\quad\text{pseudoeffective}.
\end{equation}
Choose sections $s_0,\dots,s_N$ generating $hH$ with $s_0\ne0$.  The
ratios $s_i/s_0$ lie in $F$, so their polar divisors are controlled.
Generation ensures a pulled-back section has zero order on every
prime divisor, so the divisor of $f_1^*s_0$ is bounded by the sum
of the polar divisors of the $s_i/s_0$, whose right side is
pseudoeffective.

\subsection{The invariant relative section}

Suppose given an invariant section
$\tau\in H^0(V, a p^*A-f^*B)$ with $a>0$ and $B$ a line bundle on
$Y$, such that $\tau$ has order zero on at least one component above
every prime divisor of $Y$.

\begin{proposition}\label{prop:inv-nef}
In this situation $B$ is nef; explicitly, $B$ has an invariant
semipositive metric whose weights are bounded on the smooth
connected-fiber open set.
\end{proposition}

\begin{proof}[Sketch]
Apply Lemma~\ref{lem:bounded-maximum} in its invariant form.  The
invariant relative-adjoint rank-one hypothesis for every twist
follows from the invariant version of the rank-one statement in
\S\ref{sec:approxarc}: sections of the same positive multiple of
$B$ on the generic fiber cannot have independent ratios outside $F$
by construction of $F$.  Averaging the resulting metric over the
maximal compact torus preserves invariance and semipositivity.
Nefness follows by testing on curves and the projection formula, as
in \S\ref{sec:isharedtools}.
\end{proof}

\subsection{The invariant quotient}

Choose a very ample $H$ on $Y$, and let $J=\tfrac12(kL+\epsilon H)$
be an ample rational line on a suitable birational model $S'$ of
$V$, with Seshadri constant at a very general point exceeding a
prescribed constant, a smooth positive metric of curvature $\eta$,
and a quasi-plurisubharmonic $\psi$ of neat analytic singularities
with
\[
  \eta+i\partial\bar\partial\psi\ge0,
  \quad
  \psi(w)\le\tfrac{c}{2}\log|w|^2+O(1),
\]
as in Lemma~\ref{lem:sesha}.  Averaging over the maximal compact
torus preserves positivity and makes $\psi$ invariant.

Choose $z$ a very general base point at which $g\colon S'\to Y$
is submersive and maps some $x\notin D\cup E'$.  The multiplier-ideal
quotient extension theorem of Cao--Demailly--Matsumura
\cite[Theorem~1.1]{CDM2017} then produces a nonzero invariant quotient
class.  Its representative $\tau'\in H^0(S',ap^*A-f^*B)$ has
$v$-order zero on at least one component above every prime divisor
of $Y$.  Proposition~\ref{prop:inv-nef} applies to $\tau'$ and gives
an invariant semipositive metric on $B$; in particular $B$ is nef.

\subsection{Assembly}

Apply Lemma~\ref{lem:nef-descent} to the nef base line $B$.  We
obtain a smooth projective $T$-equivariant birational model $Y'$ of
$Y$, a smooth base $Z$, and a rational $T$-equivariant descent
$\rho^*B\sim_{\Q}h^*L$ with $L$ nef and big on $Z$.

The invariant section $\tau$ of $a p^*A-f^*B$ pulls back to a
$T$-invariant section of $a p^*A-f^*\rho^*B$, hence of the pullback
of $a p^*A-f^*h^*L$.  First-jet extension of $L$ on $Z$ as in
\S\ref{sec:approxarc} then gives a nonzero $T$-invariant section of
$a p^*A$ on $Y'$.  Pushing down to $X$ gives a nonzero
$T$-invariant section of $aA=-aK_X$.

\begin{theorem}[Invariant conversion]\label{thm:invariant-conversion}
Let $X$ be smooth connected projective with smoothly semipositive
$-K_X$, and let $T$ act on $X$ preserving the natural linearization.
Suppose there exist an integer $a>0$ and invariant twisted sections
of $-aK_X+M$ for some $T$-linearized line bundle $M$, whose
restriction to a source has order zero over every base prime in the
associated model.  Then $H^0(X,r(-K_X))^T\ne0$ for some $r>0$.
\end{theorem}

\subsection{Remark: the invariant index}

Under the geometric hypotheses of Theorem~\ref{thm:invariant-conversion},
in particular $q(X)=0$, the conversion also admits a cohomological
route.  The invariant-index theorem
\cite[Theorem \textup{(Invariant anticanonical index)}]{CompanionI}
applies to $X$ and its maximal compact torus, with the natural
linearization of $A=-K_X$.  It says that the invariant index agrees
with a rational polynomial on every positive multiple of some
integer, and that this polynomial's value at zero equals the actual
invariant index at zero.  The connected torus acts trivially on
ordinary cohomology by homotopy, hence on its Hodge summands.
Therefore this zero value is $\chi(X,\OO_X)$.  If it is nonzero,
some fixed $q$ has nonzero invariant $H^q(X,mA)$ for unbounded $m$.
The compact-averaged hard Lefschetz map with coefficient $(m+1)A$
produces invariant twisted forms; the argument of
\S\ref{sec:exactweights} then reaches an invariant anticanonical
section.  In particular the hypothesis $H^q(X,\OO_X)=0$ for every
$q>0$ gives index value $1$ and $q(X)=0$, consistent with
Theorem~\ref{thm:invariant}.